REESE   LIBRARY 


UNIVERSITY  OF  CALIFORNIA, 


FIELD-MANUAL 

FOR 

RAILROAD  ENGINEERS 


BY 

J.  C.  NAGLE,  M.A.,  M.C.E., 

Professor  of  Civil  Engineering  in  the  Agricultural 
and  Mechanical  College  of  Texas. 


FIRST    EDITION. 
FIRST    THOUSAND 


NEW  YORK 

JOHN  WILEY   &   SONS. 
LONDON:    CHAPMAN  &  HALL,   LIMITED, 

1897. 


Copyright,  189? 

BY 

J.   C.   ^AGLE. 


ROBERT    DKUMMOND,    ELKCTROTYPER   AND    PRINTER,    NEW   YORK. 


PREFACE. 


EASE  of  reference  and  uniformity  of  notation  are  essential  in  a 
book  that  is  to  be  consulted  in  the  field.  With  this  in  mind  an 
effort  has  been  made  in  the  following  pages  to  secure  a  systematic 
arrangement  of  the  subject-matter  and  uniformity  of  terms  and 
notation.  Except  for  a  few  cases  Greek  letters  have  been  avoided 
and  a  single  letter  is  used  to  designate  an  angle.  In  so  far  as 
practicable  each  figure  is  intended  to  be  self-explanatory,  so  that 
the  explanations  necessary  in  connection  with  the  problems  have 
been  reduced  to  a  minimum.  Algebraic  equations  stand  each  in 
a  distinct  line,  thus  rendering  them,  more  easily  read. 

A. knowledge  of  the  elements  of  geometry  and  trigonometry  has 
been  assumed,  and  only  in  the  derivation  of  a  few  formulas  in 
connection  with  the  theory  of  transition-curves  will  any  higher 
mathematics  be  needed.  But  these  formulas  may  be  accepted  by 
the  reader  who  is  unfamiliar  with  the  calculus  without  in  any 
way  affecting  his  ability  to  understand  their  applications  or  to 
follow  subsequent  reasoning. 

One  can  most  readily  turn  to  what  he  wants  in  a  book  after  hav- 
ing become  familiar  with  its  contents  in  the  classroom.  Keeping 
this  in  mind  this  book  has  been  written  so  that  it  may  be  used  as 
a  text  as  well  as  for  reference  in  the  field.  Wherever  practicable 
solutions  to  problems  have  been  given  in  a  rigid,  general  form, 
followed  by  illustrative  examples,  so  that  the  student  need  not 
lose  sight  of  the  principle  involved  while  following  the  solution 
for  a  particular  case.  Wherever  approximate  solutions  seemed 
preferable  they  have  also  been  given  and  their  limitations  pointed 
out. 

Free  use  has  been  made  of  the  Table  of  Functions  of  a  One- 
degree  Curve,  thus  reducing  the  labor  of  field  computations.  By 
defining  the  degree  of  curve  with  reference  to  short  chords  for 


IV  PREFACE, 

sharp  curves- -and,  with  tables  of  Radii,  Long  Chords,  :\Iid 
ordinates,  etc.,  based  on  appropriate  equations — the  errors  result- 
ing from  assuming  the  radius  to  vary  inversely  with  the  degree 
of  curve  will  generally  be  found  to  be  quite  small. 

Chapter  I  gives  briefly  the  general  method  of  making  Re- 
connoissance;  Chapter  II  treats  of  Preliminary  Surveys;  while 
Chapter  III  relates  to  Location. 

Chapter  IV,  on  Transition-curves,  follows  the  method  adopted 
by  Professor  Crandall,  and  enables  one  to  locate  the  transition- 
curve  with  rigid  accuracy  where  such  is  necessary.  Approximate 
methods  are  also  given  by  means  of  which  the  curve  may  be  as 
easily  located  as  any  of  the  more  limited  easement  curves  ordi- 
narily met  with. 

Chapter  V,  on  Frogs  and  Switches,  contains  all  that  is  necessary 
for  their  location.  The  formulas  have  been  arranged  to  give  the 
desired  quantities  in  terms  of  the  frog  number  whenever  the  re- 
sulting equations  would  be  easier  of  application  than  the  trigono- 
metric ones  usually  given.  The  turnout  tables  are  unusually  full 
and  give  not  only  the  theoretical  lead  but  the  stub  lead  as  well, 
from  which  the  practical  lead  can  be  at  once  found  when  the 
length  of  switch-rail  is  known. 

Chapter  VI,  on  Construction,  tells  how  to  set  slope-stakes,  and 
gives  simple  methods  for  computing  areas  and  volumes  either 
directly  or  by  the  use  of  tables.  A  short  table  of  prismoidal 
corrections  is  given  for  end  sections  level,  and  also  a  formula  for 
three-level  sections,  by  means  of  which  a  suitable  table  may  be 
computed  if  desired. 

The  tables  at  the  end  of  this  book  have  been  arranged  with  a 
view  to  ease  of  reference,  for,  whatever  the  character  of  the  text, 
the  chief  value  of  a  field-book  must  depend  upon  the  ease  with 
which  the  tables  may  be  consulted  and  upon  their  extent  and 
accuracy.  Table  IX— Functions  of  a  One-degree  Curve  — sepa- 
rates the  logarithmic  functions  on  the  one  side  from  the  natural 
functions  on  the  other  and  will  be  of  assistance  in  locating  these 
tables.  Table  XVI — Transition-curve  Table — reading  lengthwise 
of  the  page,  likewise  serves  to  separate  the  trigonometric  tables 
from  the  miscellaneous  tables  that  follow. 

Some  engineers  object  to  the  use  of  logarithmic  tables  in  the 
field,  but  for  them  the  natural  functions  are  at  hand;  while  for 
those  who  prefer  logarithms  the  five-place  tables  of  logarithmic 
sines,  cosines,  etc.,  will  be  found  easy  to  consult  and  interpolate 
between, 


PREFACE.  V 

All  trigonometric  tables  are  five-place,  and  others  were  carried 
to  as  many  decimal  places  as  their  character  demanded. 

Tables  I,  III,  IV,  and  V  have  been  computed  to  agree  ^vith 
the  definition  of  the  degree  of  curve  requiring  curves  sharper 
than  7°  to  be  run  with  chords  less  than  100  feet  in  length,  as 
described  in  the  text.  Tables  XVII  and  XVIII  were  also  com- 
puted expressly  for  this  book. 

Tables  VI  and  XXVII  are  from  electrotypes  from  Cavhart's 
Field  Book  for  Civil  Engineers  and  were  furnished  by  Ginn  &  Co. 
Electrotypes  of  Tables  II,  X,  XII,  XIII,  XIX,  XX,  XXIV,  XXV, 
XXVI,  and  also  XVI  —  this  last  being  from  Crandall's  book, 
The  Transition  Curve — were  furnished  by  John  Wiley  &  Sons. 

Of  the  others,  some  were  arranged  from  standard  tables  and 
others  adapted  in  part  and  extended  to  increase  their  usefulness. 

It  will  be  noticed  that  vertical  lines  have  been  omitted  wher- 
ever practicable,  thus  rendering  it  easier  to  refer  to  the  tables. 

Acknowledgments  are  due  my  associate,  Professor  D.  W. 
Spence,  for  aid  in  making  the  tabular  computations  and  in  reading 
proof. 

J.  C.  NAGLE. 

COLLEGE  STATION,  TEXAS,  May,  1897. 


CONTENTS. 


CHAPTER  I. 

RECONNOISSANCE.       ^^g^lFORH^ 
ARTICLE  1.    OBJECTS  OF  RECONNOISSANCE— How  MADE. 

SECTION  PAGE 

1.  Relative  Importance  of  the  Work  of  Reconnoissance  and  Location..  1 

2.  Object  of  Reconnoissance 2 

3.  The  Instruments 2 

4.  Use  of  Maps 4 

5.  Making  the  Reconnoissauce ....   4 

CHAPTER  II. 

PRELIMINARY   SURVEYS. 

ARTICLE  2.    OBJECTS;  THE  FIELD  CORPS;  DUTIES  OF  THE  CHIEF. 

6.  Objects  of  Preliminar}-  Surveys  6 

7.  The  Exploration-line 6 

8.  Data  Sought  in  Making  Preliminary  Surveys 7 

9.  The  Field  Corps 7 

10.  The  Chief  of  Party,  Duties  of 7 

ARTICLE  3.    THE  TRANSIT  PARTY. 
A.    DUTIES  OF  THE  MEMBERS. 

11.  Composition  of  the  Transit  Party 8 

12.  The  Transitman 8 

13-17.  Other  Members  of  the  Party £ 

18.  Instruments 9 

B.     TRANSIT  ADJUSTMENTS— THE  VERNIER. 

19.  Kind  of  Transit 9 

20.  To  Adjust  the  Plate  Levels  1C 

21.  Parallax 1C 

22.  To  Adjust  the  Line  of  Collimation 1C 

23.  To  Adjust  the  Standards 11 

vii 


VI 11  CONTENTS. 


SECTION  PAGE 

24.  To  Adjust  the  Level  on  Telescope 12 

25    Direct  and  Retrograde  Verniers  . .    13 

26.  The  Least  Count  of  a  Vernier  „ Id 

27.  To  Read  a  Vernier  14 

C.  ACCESSORIES. 

(1°)  The  Gradienter. 

28.  Description  and  Method  of  Using  Gradienter 14 

(2°)  The  Stadia,  or  Telemeter. 

29.  Principle  of  the  Stadia 15 

30.  Formula  for  Line  of  Sight  Horizontal 15 

31.  Formulas  for  Line  of  Sight  Inclined 1C 

32.  The  Instrumental  Constant,  To  Find 17 

33.  Reducing  the  Notes 17 

D.  FIELD-WORK. 

34.  Station  Numbers 18 

35.  Hubs  or  Plugs 18 

36.  Reference-points    ..'..'.' 18 

37.  Alignment 18 

38.  Form  of  Transit  Notes  19 

39.  Stadia  Methods  for  Preliminary  Surveys 19 

E.      OBSTACLES  IN  TANGENT. 

41.  To  Pass  an  Obstacle  by  Means  of  Parallel  Lines 20 

42.  To  Pass  an  Obstacle  by  Angular  Deflections 20 

43.  To  Measure  across  a  River 21 

ARTICLE  4.     THE  LEVEL  PARTY. 

44.  Make-up  and  Instruments 23 

45.  Work  of  the  Leveler  23 

46.  Work  of  the  Rodman 23 

ADJUSTMENTS  OF  THE  LEVEL. 

47.  To  Adjust  the  Line  of  Collimation 23 

48.  To  Adjust  Che  Level-bubble 24 

49.  To  Adjust  the  Wyes 25 

B.      THEORY  OF  LEVELING. 

50.  True  and  Atmarent  Level 25 

51.  The  Error  Due  to  Curvature 25 

52.  The  Difference  of  Elevation  of  Two  Points 26 

C.      FIELD-WORK. 

53.  The  Datum 27 

54  Bench-marks 27 

55  Work  in  the  Field .  28 


CONTENTS.  ix 


SECTION  PAGE 

56.  The  Level  Notes 28 

57.  Precautions  when  Using  Level  29 

58.  The  Rod 29 

ARTICLE  5.    THE  TOPOGRAPHIC  PARTY. 

59.  Instruments  Used;  Area  to  be  Mapped 30 

60.  Methods  of  Recording  Data 30 

61.  Topographers'  Field-sheets 31 

62.  Use  of  the  Slope-level . .   31 

63.  Cross  section  Rods 3-J 

64.  The  Transit  and  Stadia  in  Topographical  Surveying  32 

ARTICLE  6.    PRELIMINARY  ESTIMATES.  . 

66.  Map  of  Preliminary  Lines  .   32 

67.  The  Profile     33 

68.  Preliminary  Estimates  of  Quantities 33 

G9.  Report  of  the  Locating  Engineer 34 


CHAPTER  III. 

LOCATION. 

• 

ARTICLE  7.    PROJECTING  LOCATION. 

70.  Problems  Involved  in  the  Paper  Location 35 

71.  Hints  Regarding  Methods  of  Projecting  the  Line  35 

72.  The  Curve-protractor 36 

?3.  Work  in  the  Field 37 

ARTICLE  8.    SIMPLE  CURVES. 

A.    DEFINITIONS  AND  FORMULAS. 

74.  Definitions 38 

75.  To  Find  the  Radius  R,  the  Degree  of  Curve  Being  Known 40 

76.  To  Find  the  Length  of  Curve 42 

77.  The  Functions  of  a  One-degree  Curve 42 

79.  To  Find  D,  R  and  C  Being  Known 43 

80.  To  Find  the  Tangent  Distance  T,  /  and  R  Being  Known 43 

81.  To  Find  R,  Given  I  and  T 44 

82.  Given  7  and  D,  to  Find  the  Long  Chord  L  C 44 

83.  Ordinates  from  Chord 45 

84-86.  To  Find  the  External  E 48 

87.  To  Find  R,  E  and  1  Given 49 

88.  To  Find  T,  E  and  1  Given 49 

89.  To  Find  the  Deflection  Offset  from  Chord  Produced 49 

90.  To  Find  the  Tangent  Deflection  Offset 50 

91 .  The  Sub-tangential  Deflection  Offset 51 

92.  To  Find  the  Tangent  Offset  z 52 

93.  Difference  in  Length  of  Arc  and  Long  Chord  , 53 


CONTENTS. 


E       LOCATING  SIMPLK  CURVES. 
SECTION  PAGE 

94.  To  Locate  a  Curve  with  the  Chain  by  Offsets  from  Chords  Produced  55 

St.x  To  Locate  a  Curve  by  Offsets  from  Tangent 57 

96.  To  Locate  a  Curve  by  Offsets  from  a  Long  Chord 58 

97.  To  Locate  a  Curve  with  Transit  and  Chain 59 

98.  The  Index-angle GO 

99.  Subdeflection-angles : 60 

100-101.  Transit  Notes 61 

C.      OBSTACLES. 

102.  To  Pass  an  Obstacle  on  a  Curve 63 

103.  To  Locate  a  Curve  when  the  P.  C.  is  Inaccessible 64 

104.  To  Pass  to  Tangent  when  the  P.T.  is  Inaccessible C7 

105-107.  To  Pass  a  Curve  through  a  Given  Point 69 

308.  To  Locate  a  Tangent  to  a  Curve  from  an  Outside  Point .  71 

109.  To  Run  a  Tangent  to  Two  Curves  of  Contrary  Flexure 72 

D.      CHANGE  OF  LOCATION. 

110.  To  Locate  a  Curve  Parallel  to  a  Given  Curve 73 

111.  To  Change  P.O.  in  Order  to  Make  P.T.  Fall  in  a  Parallel  Tangent. .      74 

112.  To  Change  R  and  P.O.  to  make  P.T.  Fall  in  Parallel  Tangent,  on 

Same  Radial  Line 75 

113  To  Find  Change  in  P.O.  or  R  for  a  Given  Change  in  / 76 

114.  Required  the  Change  in  P.O.  and  R  for  a  Given  Change  in  /.  the 

P.T7.  Unchanged  .  . . . ..  77 

115  To  Find  New  Radius  for  a  Given  Change  in  T 77 

116.  To  Find  New  R  to  Connect  P.  C.  with  a  Parallel  Tangent 78 

ARTICLE  9.    COMPOUND  CURVES. 
A.    LOCATION  PROBLEMS. 

117.  Given  Both  Tangents  and  One  Radius,  to  Find  the  Other  Radius  ...    80 

118.  Given  One  Radius,  the  Long  Chord  and  the  Angles  it  Makes  with 

Tangents,  to  Find  the  Other  Radius  and  Central  Angles. ...     82 

119.  Given  the  Radii  and  Central  Angles,  to  Find  the  Tangents,  the  Long 

Chord .  and  the  Angles  it  Makes  with  Tangents 80 

120.  Given  the  Long  Chord  and  Angles  Made  with  Tangents,  to  Find 

Both  Radii  when  Common  Tangent  is  Parallel  to  Long  Chord 83 

B.      OBSTACLES. 

121.  To  Locate  Second  Branch  when  P.  C.  is  Inaccessible 84 

C.     CHANGE  OF  LOCATION. 

122.  To  Compound  a  Simple  Curve  so  P.T.  shall  Fall  in  a  Parallel  Tan- 

gent      85 

123.  To    Find  Change  in  P.C.C.  Necessary  to  Make  P.T.  Fall  in  a  Par- 

allel  Tangent  86 

124.  To  Change  P.C.C.  and  Second  Radius  so  P.T.  shall  Fall  in  a  Par- 

allel Tangent,  on  Same  Radial  Line 89 


CONTENTS.  XI 


SECTION  PAGE 

125.  To  Change  P.C.C.  and  Second  Radius  to  Cause  P.T.  to  Fall  at  a 

New  Point  in  Same  Tangent 9 ! 

120.  To  Substitute  a  Three-centered  Compound  Curve  for  a  Simple  One.  94 

127.  To  Substitute  a  Curve  for  a  Tangent  Uniting  Two  Curves 95 

ARTICLE  10.    TRACK  PROBLEMS. 

128.  Reversed  Curves,  Where  to  Use 9G 

129.  To  Connect  a  Located  Curve  with  an  Intersecting  Tangent 97 

130   To  Locate  a  Y 100 

131 .  A  Reversed  Curve  between  Parallel  Tangents  102 

132.  A  Crossover  between  Parallel  Tracks  when  a  Fixed  Length  of  Tan 

gent  is  Inserted 105 

133.  A  Reversed  Curve  with  Unequal  Angles 106 

134.  A  Reversed  Curve  between  Fixed  Points 106 

135.  To  Connect  Two  Divergent  Tangents  by  a  Reversed  Curve . . . .  107 

136.  To  Change  P.R.C  so  P.T.  shall  Fall  in  a  Parallel  Tangent 108 

137   To  Find  the  Radius  of  a  Curved  Track    .  . .  109 


CHAPTER  IV. 

TRANSITION-CURVES. 

ARTICLE  11.    THEORY  OF  THE  TRANSITION-CURVE. 

138.  Elevation  of  Outer  Rail  on  Curves 110 

139    Requirements  of  the  True  Transition-curve Ill 

1-10.  Notation  Employed ill 

141.  Equation  of  Transition-curve  .....  112 

142.  Transition-curve  Angle,  / , .    .  114 

143.  Coordinates  of  Points 114 

144.  Deflection-angles 115 

145.  Explanation  of  Transition-curve  Tables.  118 

146   To   Unite  the  Branches  of  a  Compound  Curve  by  a  Transition- 
curve , . . . .  119 

147.  Length  of  Transition-curve  to  be  Taken 121 

ARTICLE  12.    FIELD- WORK. 

A.      FIELD    FORMULAS. 

148.  When  to  Use  the  Simplified  Formulas    122 

149.  Simplified  Formulas  for  Transition-curves 122 

150.  Offsets 124 

151.  Compound  Curves 125 

B.      SETTING  OUT  TRANSITION-CURVES. 

153.  Location  by  Offsets 125 

154.  Location  by  Deflection -angles  ...    126 

155.  Form  of  Transit  Notes  for  Transition  curves. .  128 


Xll  CONTENTS. 


ARTICLE  13.    TRANSITION  CURVE  PROBLEMS. 
SECTION  PAGE 

156.  Tangent  Distances  and  External  for  Equal  Offsets 109 

157.  Tangent  Distances,  Offsets  Unequal 130 

158.  Transition-curves  Inserted  without  Changing  the  Vertex  of  Cir 

cular  Curve ...  131 

159.  Transition-curves  Inserted  with  Least  Deviation  from  Old  Track... .  133 

160.  Transition-curves  Inserted   at  Ends  of  Long  Circular  Curve,  Cen- 

tral Portion  Undisturbed  133 

161.  Transition-curve  Inserted  at  P.C.C.  by  Changing  Radius  of  Second 

Branch   136 

162.  To  Insert  Transition-curves  at  the  Ends  of  Two  Circular  Curves 

United  by  a  Common  Tangent 138 

163.  To  Unite  a  Tangent  and  Circular  Curve  when  the  Offset  Cannot  be 

Directly  Measured  139 

164.  Inserting  Transition-curves  in  Old  Track 140 

165.  Remarks  on  Tabular  Interpolations  140 


CHAPTER  V. 

FROGS  AND  SWITCHES. 

ARTICLE  14.    TURNOUTS. 

A.    TURNOUTS  FROM  STRAIGHT  LINES.^ 

166.  Definitions 143 

167.  To  Find  the  Lead,  Z,  and  Radius,  R,  in  Terms  of  the  Frog  Number, 

N,  and  Gauge,  g 144 

168.  Given  R  and  g,  to  Find  N,  J,  and  Frog-angle,  F 146 

169.  To  Find  Theoretic  Length  of  Switch-rail 146 

170.  To  Fiud  Lead  and  Number  of  Crotch-frog  for  a  Double  Turnout  to 

Opposite  Sides  of  Main  Track 147 

171.  To  Find  Turnout  Radius  and   Lead  of  Crotch -frog  in  Terms  of 

Crotch-frog  Number 148 

172.  To  Find  Radius  of  Curve  from  Point  of  Middle  Frog  to  Point  of 

Main  Frog,  Given  _ZV,,  N.  and  N' 148 

173.  Double  Turnout  to  Same  Side  of  Main  Track 150 

174.  To  Find  Radius  of  Curve  between  Frog-points  for  a  Double  Turn- 

out to  Same  Side  of  Main  Track .  151 

175.  To  Unite  Main  Track  with  Siding.  Reversing  Point  Opposite  Frog  . .  152 

176.  To  Lay  Out  a  Ladder-track  153 

B.      TURNOUTS  FROM  CURVES. 

177.  To  Find  Lead  and  Radius  for  Turnout  to  Concave  Side  of  Main 

Line       .  ir>l 

178.  To  Find  Lead  and  Radius  Turnout  to  Convex  Side 157 

179.  To  Find  Theoretic  Length  of  Switch-rail !.r>S 

180.  To  Unite  Main  Track  with  a.  Concentric  Siding    .   .  160 


CONTENTS.  Xlll 

C.      THE   riTUB   LEAD. 
SECTION  PAGE 

181.  Definitions 1W 

18-2.  Given  N,  t,  and  g,  to  Find  the  Stub  Lead 162 

183.  Turnout  Table  and  Explanation , . .   163 

184.  To  Stake  Out  a  Turnout 16o 

185.  Curving  Rails 165 

ARTICLE  15.    CROSSOVERS. 

186.  Crossover  between  Parallel  Straight  Tracks,  a  Tangent  between 

Frog-points 166 

187.  A  Crossover  in  the  Form  of  a  Reversed  Curve 168 

188.  A  Crossover  with  Fixed  Length  of  Intermediate  Tangent 168 

189.  A  Crossover  between  Curved  Main  Tracks 168 

ARTICLE  16.    CROSSING-FROGS  AND  CROSSING-SLIPS. 

A.  CROSSING-FROGS. 

191.  Length  of  Rail   Intercepted   between    Two  Intersecting   Straight 

Tracks.     -. 170 

192.  Angles  of  a  Set  of  Crossing  frogs,  One  Track  Curved 170 

193.  Angles  of  a  Set  of  Crossing-frogs,  Both  Tracks  Curved 171 

B.  CROSSING-SLIPS. 

195.  Length  and  Radii  of  Slip-rails,  Both  Tracks  Straight 172 

106.  Length  and  Radii  of  Slip-rails,  One  Track  Curved  172 

197.  Length  and  Radii  of  Slip-rails,  Both  Tracks  Curved 173 


CHAPTER  VI. 

CONSTRUCTION. 

ARTICLE  17.    DEFINITIONS  :  GENERAL  CONSIDERATIONS  ;  VERTICAL 
CURVES  ;  ELEVATION  OF  OUTER  RAIL. 

199.  The  Division  Engineer 176 

200.  The  Resident  Engineer 176 

201  -204.  Definitions 177 

205.  To  Find  che  Grade-point,  Longitudinal  Slope  Uniform 178 

206   Vertical  Curves 178 

207.  Elevation  of  Outer  Rail  on  Curves 182 

208.  Easing  Grade  on  Curves , 183 

ARTICLE  18.    EARTH-WORK. 
A.    SETTING  SLOPE-STAKES. 

209.  The  Distance  Out  for  Level  Sections I8a 

210.  To  Find  Position  of  Slope-stakes  for  Surface  Inclined 184 

211.  Cross -section  Notes 186 

212.  Irregular  Sections 187 

U3.  Staking  Out  Openings - 187 


XIV  CONTESTS. 


SECTION  PAGE 

214.  Manuer  of  Marking  Stakes is? 

215.  Shrinkage— Growth j,-7 

216.  Borrow-pits,  Drainage  of ,  etc 188 

B.      AREAS  OF  SECTIONS. 

218.  Area  of  Three-level  Section 188 

219.  Area  of  Five-level  Section 189 

220.  General  Formula  for  Areas 190 

221.  Explanation  of  Table  of  Areas  of  Level  Sections  and  the  Three- 

level  Correction 191 

C.      VOLUME  OF  EARTHWORK. 

222.  Where  Cross-sections  should  be  Taken 193 

228.  Volume  by  Averaging  End  Areas. . .  19i> 

224.  The  Prismoidal  Formula . .   J93 

225.  Form  of  Record 195 

226.  The  Prismoidal  Correction 195 

227.  Computation  of  Volumes  when  Passing  from  Cut  to  Fill      198 

228.  Use  of  Tables  of  Volumes  in  Making  Preliminary  Estimates 199 

229    Side  Ditches 199 

230.  Earthwork  on  Curves 199 

231.  Overhaul '. 201 

ARTICLE  19.    GRADE  AND  BALLAST  STAKES,  CULVERTS,  BRIDGES, 
AND  TUNNELS. 

232.  Grade  and  Center  Stakes 202 

233.  Ballast-stakes 202 

235.  Openings  of,  for  Culverts,  Trestles,  etc 202 

236.  Bridge  Piers  and  Abutments .203 

237.  Tunnels 204 

ARTICLE  20.    MONTHLY  AND  FINAL  ESTIMATES. 

238.  Monthly  Estimates 205 

239.  Measurements  for  Earthwork 206 

240.  Classification  of  Earthwork 206 

211.  The  Progress  Profile 207 

242.  Masonry  Estimates 207 

243.  Bridge  Estimates 207 

244.  Track  Material 207 

245.  Blank  Estimate  Sheets 208 

246.  Monthly  Payments 208 

247.  Extras 208 

248.  Final  Estimate 208 

249.  Acceptance 209 

TABLES. 

Table  Showing  Length  of  Transition-curve  to  be  Taken  12i 

Table  of  Values  of  g  -  Vgi  for  Stub  Lead 163 

Turnout  Table 164 

Table  of  Corrections  for  Vertical  Curves 181 


CONTENTS.  XV 


PAG  E 

Table  of  Elevation  of  Outer  Rail  on  Curves  182 

Table  of  Prlsmoidal  Corrections  for  Level  Sections 196 

I.  Radii  of  Curves 212 

II.  Minutes  in  Decimals  of  a  Degree 215 

III.  Tangential  Offsets  216 

IV.  Long  Chords  and  Actual  A-rcs 217 

V.  Mid-ordinates  to  Long  Chords 218 

VI.  Logarithms  of  Numbers , 220 

VII.  Logarithmic  Sines  and  Cosines. 838 

VIII.  Logarithmic  Tangents  and  Cotangents 253 

IX.  Functions  of  a  One-degree  Curve 268 

X.  Natural  Sines  and  Cosines  298 

XI.  Natural  Secants  and  Cosecants 307 

XII.  Natural  Tangents  and  Cotangents 320 

XIII.  Natural  Versines  and  Exsecants 332 

XIV.  Coordinates  for  Transition-curves 355 

XV    Deflection-angles  for  Transition-curves 356 

XVI.  Transition-curve  Table 358 

XVII.  Areas  of  Level  Sections 371 

XVIII.  Corrections  for  Three-level  Ground 375 

XIX.  Cubic  Yards  per  100  ft.  in  Terms  of  Center  Height 376 

XX.  Cubic  Yards  per  100  ft.  in  Terms  of  Sectional  Area. ...  382 

XXI.  Rise  per  Mile  of  Various  Grades 386 

XXII.  Slopes  for  Topography 387 

XXIII.  Material  Required  for  One  Mile  of  Track . .  387 

KXIV.  Mutual  Conversion  of  Feet  and  Inches  into  Meters  and  Centi- 
meters   388 

XXV.  Mutual  Conversion  of  Miles  and  Kilometers 389 

XXVI.  Length  of  1'  Arc  of  Latitude  and  Longitude 389 

XXVII.  Trigonometric  and  Miscellaneous  Formulas  390 


A    FIELD-MANUAL    FOE    RAILROAD 
ENGINEERS. 


CHAPTER  I. 

RECONN01SSANCE. 

ARTICLE  1.     OBJECTS  OF  RECONNOISSANCE — How  MADE. 

1.  THE  question  of  the  selection  of  the  proper  route  for  a  line 
of  railway  is  essentially  an  economic  one.  involving  not  only  the 
cost  of  construction,*  but  of  maintenance  and  operation,  and  a 
consideration  of  the  immediate  and  future  traffic  likely  to  pass 
over  the  completed  road. 

The  engineer  upon  whom  devolves  the  duty  of  making  the 
surveys  for  a  railroad  is  not  often  called  upon  to  determine 
whether  it  should  or  should  not  be  built,  though  his  preliminary 
estimate  may  decide  those  whose  duty  it  is  to  do  so  :  the  problem 
confronting  him  is  liow  to  secure  the  best  line,  answering  a  given 
purpose,  for  the  least  cost.  Keeping  in  mind  the  proper  working 
of  the  completed  road,  the  problem  may  be  divided  into  two  gen- 
eral parts  : 

First.  The  selection  of  the  general  route  between  terminal 
points,  and  in  some  cases  the  selection  of  the  terminals  them- 
selves. 

Second.  The  fitting  of  the  line  to  the  ground  in  such  a  manner 
as  will  render  the  cost  of  constructing  and  operating  the  road  a 
minimum. 

The  first  is  by  far  the  more  important  and  difficult  operation, 
requiring  the  highest  grade  of  engineering  skill — a  fact  too  sel- 
dom recognized  by  those  selecting  engineers  for  this  work.  The 
acquirement  of  the  necessary  skill  can  result  only  from  long 
practice  and  close  observation,  coupled  with  the  ability  to  fully 


2  A    FIELD-MANUAL   FOR    RAILROAD    KN(JI  N  KEIIS. 

grasp  and  weigh  till  the  complex  features  of  the  question.  A 
passing  reference  only  can  be  made  to  it  in  this  little  volume, 
which  is  intended  to  furnish  hints  and  aids  to  the  better  execution 
of  the  second  part.  For  the  benefit  of  the  beginner  who  has  to 
do  with  the  location  and  construction  a  few  definitions  and  hints 
relating  to  reconuoissance  will  be  given  before  going  on  to  the 
special  problems  arising  in  the  work  of  the  railroad  engineer. 

2.  The  Reconnoissance  is  a  rapid,  general  survey  of  the  area 
through  which  the  proposed  railroad  must  pass,  made  only  with 
such  instruments  as  can  be  easily  carried,  and  which  should  ena- 
ble the  engineer  to  restrict  the  more  accurate  instrumental  work 
that  follows  to  one  or  two  general  lines.     The  time  required  for 
this  part  of  the  work  will  in  general  be  only  a  small  fraction  of 
the  time  consumed  in  location,  involving  the  service  of  very  few 
men;  yet  there  is  no  part  of  the  work  more  rapidly  and    im- 
properly done — not  always  because  the  engineer  in  charge  under- 
estimates its  importance,  but  because  he  is  not  usualty  allowed 
sufficient  time  in  which  to  study  thoroughly  the  area  under  con- 
sideration. 

Properly  the  reconnoissance  includes  the  determination  of  the 
terminal  points  of  the  road,  but  the  locating  engineer  is  usually 
relieved  from  the  necessity  of  selecting  these  points,  and  the 
question  reduces  to  that  of  finding  the  best  available  line  which 
admits  of  being  built,  maintained,  and  operated  at  the  least  cost 
between  two  given  points. 

The  reconnoissance  must  be  made  over  an  area — not  a  line  or 
lines.  Even  what  seems  the  most  unpromising  portion  should 
be  carefully  studied,  for  the  engineer  can  never  be  satisfied  he 
has  selected  the  best  route  until  he  has  convinced  himself  by  care- 
ful study  that  all  others  are  inferior.  Too  much  haste  on  recon- 
noissance means  either  a  poor  line  or  a  much  greater  expenditure 
of  time  and  money  on  the  preliminary.  No  amount  of  notes  or 
topography  can  take  the  place  of  an  intimate  personal  knowledge 
of  the  problems  to  be  encountered,  and  hence  the  reconuoissance 
and  preliminary  survey  should  be  made  by  the  engineer  who  is 
to  locate  the  road. 

3.  The  Instruments  needed  will  rarely  be  more  than  a  pocket- 
compass,  hand-level,  aneroid  barometer,  field-glasses,  and  some- 
times a  pedometer  or  an  odometer. 

(a)  The  Pocket-compass  is  used  to  obtain  the  magnetic  bear- 
ings of  lines  and  the  angles  they  make  with  each  other. 


RECONNAISSANCE.  3 

(6)  The  Hand-level  enables  one  to  obtain  differences  of  ele- 
vation between  points  not  fur  apart. 

(c)  The  Aneroid  Barometer  gives  approximate  heights  of  the 
mercury  column,  and  serves  to  roughly  determine  the  difference 
of  elevation  of  given  points.  In  addition  to  the  scale  giving 
readings  in  inches,  it  should  have  also  a  scale  graduated  to  give 
readings  in  feet.  If  two  aneroids,  which  have  been  previously 
compared,  are  read  simultaneously,  one  at  each  of  the  points 
whose  difference  of  elevation  is  desired,  or  if  the  same  aneroid  is 
read  at  each  successively  at  a  short  interval  of  time,  during  which 
the  atmospheric  pressure  has  not  sensibly  altered,  we  may  find 
the  difference  of  elevation  by  the  formula* 


d  =  60000  (log  H-  log  h)l  +  9     .     .    (1) 

in  which  d  is  the.  difference  of  altitude  in  feet,  H  and  h  the 
barometric  readings  in  inches  —  the  logarithms  being  of  the  com- 
mon or  Briggs  kind,  I7  and  t  the  temperatures  of  the  two  stations 
in  Fahrenheit  degrees. 

If  the  sum  of  the  temperatures,  T  -\-  1,  is  taken  as  105°,  formula 
(1)  reduces  to 

d-  63000  (log  H-  log  A)  ......     (!') 

EXAMPLE.  —  The  reading  of  the  barometer  at  the  foot  of  a 
mountain  is  28.8  inches,  and  at  the  top  26.7  inches.  Required 
the  height  of  the  mountain. 

By  (!').  d  =  63000  (log  28.7  -  log  26.7)  =  2071  feet. 

The  effect  of  temperature  on  the  metal  of  the  instrument 
should  be  considered  in  the  barometric  formula  when  very  pre- 
cise work  is  to  be  done  ;  but  this  correction,  being  small,  may  be 
neglected  in  the  rough  work  of  reconnoissance,  particularly  since 
the  makers  of  the  instrument  construct  it  in  such  a  way  as  to 
compensate,  as  closely  as  possible,  for  such  changes  of  tem- 
perature. 

(d)  The  Pedometer  is  an  instrument  which  automatically 
counts  the  number  of  steps  made  by  a  person  when  the  instru- 
ment is  attached  to  his  belt  ;  then,  knowing  the  average  length 
of  step,  the  distance  passed  over  can  be  readily  computed. 

The  Odometer  registers  the  number  of  revolutions  of  a  wheel 
to  which  it  is  attached,  and  the  number  of  revolutions  multiplied 
by  the  circumference  of  the  wheel  gives  the  space  passed  over. 

*  See  Plymtou's  Aneroid  Barometer,  p.  38,  for  formula  (1). 


4  A   FlELr-ilANUAL    FOR   RAILROAD    ENGINEERS. 

4.  The  Map. — Before  beginning  the  reconnoissance  the  engi- 
neer should  provide  himself  with  the  best  available  map  of  the 
region  to  be  traversed  ;  if  this  is  a  topographic  one,  he  can  at 
once  determine  from  it  the  lines  that  are  likely  to  justify  an 
examination  ;  and  even  if  it  is  only  a  sketch-map,  he  can  get 
material  assistance  by  observing  the  courses  of  the  streams  and 
remembering  that  their  positions  indicate  the  relative  elevations 
of  the  portion  of  the  region  through  which  they  flow.  Thus  the 
large  streams  follow  the  lines  of  least  elevation,  and  the  manner 
in  which  the  lateral  streams  unite  with  the  principal  one  indi- 
cates the  general  trend  of  the  terrain.  Two  streams  flowing 
nearly  parallel  approach  or  recede  from  each  other  according  us 
the  intervening  land  diminishes  or  increases  in  altitude.  Two 
streams  flowing  away  from  each  other  on  opposite  sides  of  a 
divide,  and  having  their  source  therein,  approach  each  other 
closest  at  the  point  of  least  elevation,  and  indicate  the  position  of 
a  pass  or  the  lowest  point  of  the  dividing  ridge.  The  study  of 
any  good  contour  map  covering  sufficient  area  will  illustrate  the 
laws  governing  the  courses  followed  by  streams. 

The  elevations  of  a  few  correctly  mapped  points,  when  obtain- 
able, from  the  map  or  otherwise,  serve  as  a  guide  in  tentatively 
fixing  on  the  maximum  gradient  to  be  employed  and  the  amount 
of  development  needed. 

A  skillful  engineer  will  thus  be  enabled  to  project  his  lines 
with  sufficient  accuracy  to  enable  him  to  select  on  the  ground  the 
most  feasible  route  or  routes  for  his  preliminaries  in  the  least 
possible  time.  He  should  guard  against  the  conviction,  however, 
that  it  is  unnecessary  for  him  to  look  elsewhere  than  along  the 
projected  routes  ;  for  the  inaccuracies  of  the  map,  local  peculiari- 
ties, the  nature  of  the  excavation  and  embankment,  the  number 
and  cost  of  bridges  and  other  mechanical  structures, — all  these 
may  conspire  to  make  the  most  promising  map-line  inferior  to 
some  other  whose  advantages  have  to  be  sought  for  on  the 
ground. 

6.  Having  tentatively  decided  on  the  limiting  grades  and  cur- 
vature to  be  employed,  the  engineer  goes  carefully  over  the 
ground,  examining  the  entire  area  that  seems  likely  to  afford 
passage,  in  order  to  determine  whether  a  suitable  line  may  be 
secured  for  the  grades  and  curves  previously  assumed.  With  his 
pocket-compass  he  takes  the  bearings  of  lines,  and  by  means  of 
the  hand-level  and  aneroid  determines  differences  of  elevation.- 


RECONNOISSANCE.  5 

Distances  are  estimated  by  the  eye,  paced,  and  the  count  taken 
from  the  pedometer,  or,  if  the  country  admits  of  the  use  of  a 
vehicle,  taken  from  the  odometer  readings.  If  a  well-gaited 
saddle-horse  is  used,  very  good  results  may  be  gotten  by  timing 
him,  or  by  the  use  of  the  pedometer  if  his  stride  is  uniform. 

But  in  all  cases  much  dependence  must  be  placed  on  the  ability 
to  estimate  with  the  eye  differences  of  elevation  and  distances. 
The  ability  to  do  this  with  even  reasonable  accuracy  comes  only 
from  long  practice  and  careful  observation,  even  to  the  most 
gifted  in  this  respect.  New  and  unexpected  conditions  some- 
times deceive  even  the  most  practiced  eye,  but  under  ordinary 
conditions  almost  any  one  can  train  his  eye  to  estimate  horizontal 
distances  fairly  well.  Vertical  heights  are  more  deceptive,  pos- 
sibly because  we  have  less  practice  in  this  line,  and  the  mind 
seems  naturally  to  exaggerate  the  vertical  as  compared  with  the 
horizontal  ;  practice,  however,  will  enable  us  to  make  allowance 
for  the  natural  tendency  to  overestimate  heights  and  slopes. 

The  ground  should  be  gone  over  in  both  directions,  for  the  ap- 
pearance may  be  quite  different  when  approached  from  different 
quarters.  Ruling  points,  such  as  a  pass  in  the  mountains,  the 
crossing  of  a  large  stream,  or  a  town  or  city  through  which  the 
road  must  be  built,  serve  to  reduce  the  problem  to  a  number  of 
special  ones,  each  having  its  own  solution. 

In  a  mountainous  region  offering  a  limited  number  of  possible 
routes,  but  heavy  construction  work,  it  may  often  happen  that 
the  location  of  a  line  is  a  much  less  difficult  operation  than  in  an 
open,  rolling  country  offering  a  score  of  possible  lines,  between 
which  the  engineer  making  the  reconnoissance  must  decide, 
selecting  only  those  that  in  his  judgment  seem  to  justify  an 
accurate  instrumental  survey. 

The  engineer  must  keep  constantly  in  mind  all  the  factors  of 
the  general  problem  of  economic  location  and  maintenance,  and 
successful  operation  of  trains.  One  line  may  cost  more  for  con- 
struction and  maintenance  than  another,  but  less  for  operation, 
or  may  invite  less  traffic.  In  all  cases,  however,  the  question 
of  grades,  curvature,  length  of  line,  earthwork,  and  mechanical 
structures  are  the  controlling  elements  to  be  considered. 

Having  decided  upon  the  route  or  routes  over  which  to  run 
preliminaries,  these  are  marked  on  the  map,  and  the  engineering 
party  organized  and  put  in  the  field,  with  all  the  necessary 
instruments. 


CHAPTER  II. 

PRELIMINARY  SURVEYS. 

ARTICLE  2.    OBJECTS;  THE  FIELD  CORPS  ;  DUTIES  OF  THE  CHIEF. 

6.  The  Objects  of  the  preliminary  surveys  are  to  secure  all  the 
data  necessary  to  determine  which  one  of  the  routes  selected  on 
reconnoissance  is  the  most  feasible,  all  things  considered,  and  the 
approximate  cost  of  construction.     In  rough  country  it  will  be 
economical  to  make  two,  or  even  three,  surveys  over  the  route  se- 
lected for  location  before  beginning  to  place  the  line  in  the  position 
it  is  finally  to  occupy.     The  first  of  these  is  often  omitted,  and  is 
called  an  "exploration-line  "  ;  it  will  frequently  save  the  making 
of  the  more  expensive  "preliminary"  over  one  or  more  of  the 
routes. 

7.  The  Exploration-line  may  be  made  with  either  transit  or 
compass,  and  consists  of  a  rapidly  ran  line,  made  for  the  purpose 
of  determining  the  maximum  curvature  and  gradients  with  which 
to  project  the  preliminary.     It  will  not  be  necessary  to  make  a 
detailed  study  of  the  region  at  this  time,  the  distances  and  eleva- 
tions, with  such  sketch  topography  as  may  be  easily  taken,  being 
all  that  is  needed.     The  magnetic  bearing  of  lines  is  taken  by 
the  cornpassman,  and  the  chainmen  align  each  other  with  the  flag 
set  by  the  flagman.     As  the  progress  of  the  level  party  will  be 
slower  than  that  of  the  compass  party,  it  will  be  economical  to  add 
an  extra  rodman,  and  sometimes  a  recorder.     The  compassman 
may  sketch  in  the  features  adjacent  to  the  line  while  waiting  for 
his  chainmen,  who  may  be  either  in  front   of  or  behind  the  com- 
pass. 

The  stadia  method  of  surveying — to  be  spoken  of  later — would 
seem  to  offer  exceptional  advantages  for  this  work — only  three  or 
four  men  being  needed  in  addition  to  the  chief.  With  it,  by  sct^ 
ting  the  transit  over  alternate  stations,  very  rapid  progress  may  be 
made,  and  obstacles  avoided  with  as  much  or  greater  ease  than 
with  the  compass. 

The  exploration-line  will  more  than  pay  for  itself  in  showing 

6 


PRELIMINARY   SURVEYS.  V 

what  routes  it  will  be  unnecessary  to  make  preliminaries  over, 
and  in  indicating  the  most  feasible  one.  It  should  be  run  over  all 
the  routes  selected  on  reconnoissance. 

8.  The  Preliminary  Survey  follows  the  exploration,  or,  when 
this  is  omitted,  comes  next  after  the  reconnoissance.     It  may,  with 
advantage,  be  made  in  two  parts — first  and  second  preliminary. 
It  is  made  with  such  instrumental  accuracy  as   the  nature  of  the 
case  may  demand,  sufficient  data  being  obtained  to  determine  the 
best  line  on  which  to  locate  and  the  approximate  cost  of  construc- 
tion.    The  rapidity  with  which  this  work  can  be  done  will'depend 
on  the  care  with  which  the  reconnoissance  was  made.     The  pre- 
liminary line  should  approximate,  as  closely  as  the  eye  can  deter- 
mine, to  the  position  the  located  line  should  occupy,  and  forms  the 
base  on  which  the  topographic  work  rests.     In  reasonably  easy 
country,  where  exploration-lines  have  been  run,  one  preliminary 
should  suffice  for  each  route,  but  in  difficult  regions  it  will  be  best 
to  run  a  second  preliminary.     If  portions  of  the  route  are  easy,  fol- 
lowed by  difficult  parts,  it  will  often  be  sufficient  to  "  back  up  " 
and  re-run  the  difficult  portion  until  a  reasonably  satisfactory  line 
has  been  obtained. 

9.  The  Field  Corps  consists  of  a  chief  of  party,  transitman, 
leveler,    rodman,   two   chainrnen,    rear   rodman  or   "back-flag," 
stakeman,  and  two  or  more  axemen.     If  a  topographic  party  is 
added,  as  it  should  be  in  any  but  the  easiest  country,  there  will  be 
also  a   topographer  with  two  or  more   assistants.     A. cook  and 
teamster  will  be  needed  with  the  camp  outfit. 

The  corps  is  usually  divided  into  the  following  parties  : 

(a)  THE  TRANSIT  PARTY. 

(6)  THE  LEVEL  PARTY. 

(c)  THE  TOPOGRAPHIC  PARTY. 

10.  The  Chief  of  Party  receives  his  orders  from  the  chief  en- 
gineer, or  such  other  officer  as  may  be  in  charge,  directs  the  mo- 
tions of  the  surveying  corps,  and  is  responsible  for  their  conduct 
and  progress.    He  provides  accommodations  and  supplies,  pays  all 
expenses,  taking  receipts  or  vouchers  for  all  outlays — in  dupli- 
cate when  required.     In  the  less  thickly  populated  sections  he 
must  provide  tents,  wagons,  cook,  and  all  necessary  camping  outfit 
and  supplies.     He  must  direct  the  field  operations  in  person,  keep- 
ing in  advance  of  the  transit,  establish  turning-points  or  hubs, 
and  direct  the  transitman  in  the  proper  course.     He  should  keep 


8          A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

a  record — or  direct  the  transitman  and  topographer  to  do  so — of 
the  character  of  earthwork  likely  to  be  encountered,  the  places 
where  drains,  culverts,  bridges,  cattle-guards,  etc.,  are  needed; 
the  nature  of  material  for  embankment,  piling,  etc.,  adjacent  to 
the  line  ;  the  probable  amount  of  clearing  and  grubbing,  and  all 
other  features  likely  to  affect  the  cost  of  construction.  He  should 
see  that  the  names  of  property  owners  and  residents  along  the 
line  and  the  positions  and  bearings  of  property  lines,  when 
possible,  are  noted. 

He  should  have  authority  to  discharge  assistants — except  transit- 
man, leveler,  and  topographer — whose  services  are  unsatisfactory, 
and  in  many  cases  it  will  be  best  for  him  to  have  entire  control, 
engaging  or  discharging  any  member  of  the  corps  as  circumstances 
may  require. 

ARTICLE  3.     THE  TRANSIT  PARTY. 
A.  Duties  of  the  Members. 

11.  The  Transit  Party  should  consist  of  a  transitman,  head 
chainman,  rear  chaininan,  rear  flagman,  stakeman,  and  as  many 
axemen  as  may  be  required — rarely  less  than  two  even  for  open 
country. 

12.  The  Transitman  cares  for  his  instrument,  keeping  it  in  ad- 
justment; directs  the  chainmen  into  line;  notes  the  angle  between 
successive  tangents  as  read  on  plates;  notes  also  the  bearings  of 
tangents,  of  highways,  streams,  and  property  lines  (on  location), 
with  the  plus  at  which  the  line  crosses  them.     If  there  is  no 
topographic  party  he  must  make  sketches,  on  the  right-hand  page 
of  note-book,  of   the  surface  features  adjacent  to  the  line;   the 
red  line  down  the  middle  of   page   represents   the   transit   line, 
whether  straight,   broken,  or  curved,  to  which  the  sketches  are 
adjusted.     He  must  see  that  the  axemen  keep  in  line,  in  order 
that  no  unnecessary  chopping  may  be  done.     Large  trees  need 
rarely  be  felled  on  preliminary,  even  when  a  given  general  course 
has  to  be  followed,  for  small  angles  may  be  turned  to  avoid  them, 
the  deflections  to  right  being  made  to  approximately  balance  those 
to  left. 

When  the  chief  of  party  is  absent  the  transitman  is  ranking 
man,  and  will  take  temporary  charge. 

13.  The  Head  Chainman  carries  a  range-pole  or  "  flag,"  and 
drags  the  chain,  which  he  must  see  is  straight  and  horizontal 


PRELIMINARY    SURVEYS.  9 

when  setting  a  point  for  a  stake.  He  directs  the  stakeman  where 
to  drive  his  stake,  calling  out  the  number  after  the  rear  chainman 
has  read  and  called  out  the  number  on  his  stake;  he  keeps  the 
axemen  in  line  by  setting  his  flag  and  going  ahead,  directing  them 
where  to  cut  by  keeping  them  in  line  with  the  flag  and  transit. 
The  speed  of  the  party  is  dependent  on  the  rapidity  and  accuracy 
with  which  he  can  set  his  flag  in  position,  by  ranging  with  stakes 
already  set  between  him  and  transit,  and  in  seeing  that  the 
axemen  make  all  their  work  count. 

14.  The  Rear  Chainman  must  be  careful  to  hold  his  end  of 
the  chain  in  the  proper  place,  and  that  it  is  kept  straight  and  taut 
when  the  head  chainman  is  setting  a  stake.     He  must  give  all 
pluses,  note  the  number  on  each  stake  as  he  comes  up  to  it,  and 
see  that  the  stakeman  has  marked  it  correctly;  he  must  make  a 
note  of  pluses  for  roads,  fences,  streams,  etc.,  to  be  given  to  the 
transitman  later  on. 

15.  The   Stakeman  must  keep  himself   supplied  with   stakes 
about  li"  X  2"  X  24",  marking  the  number  on  them  plainly,  and 
driving  them  as  directed  by  the  head  chainman. 

If  sawed  stakes  are  not  provided,  he  must  cut  the  stakes  and 
face  them  for  the  numbers.  He  must  keep  on  hand  a  number  of 
plugs  or  "hubs,"  to  be  driven  flush  with  the  ground  and  having 
the  point  where  flag  rested  marked  with  a  tack.  About  ten  or 
twelve  inches  to  the  left  of  and  facing  the  hub  a  guard  stake  is 
driven,  on  which  is  marked  the  station  number,  and  which  enables 
one  to  find  the  hub  at  any  time. 

16.  The  Axemen  do  all  necessary  clearing  and  chopping  in 
order  that  the  transit  and  level  parties  may  have  a  clear  sightway, 
and  yet  restrict  the  work  of  clearing  to  a  minimum.     One  of  them 
may  be  detailed  to  keep  the  stakeman  supplied  with  stakes. 

17.  The  Rear  Flagman  holds  his   flag  on  the   last  turning- 
point  for  the  transitman  to  use  in  back-sighting. 

18.  The  Instruments   used   by  the   party  are  the  transit   (or 
compass),  one-hundred-foot  chain  or  tape,  range-poles,  and  the 
necessary  axes  and  hatchet  for  axemen  and  stakeman. 

B.    Transit  Adjustments — The  Vernier. 

19.  For  railroad  work  the  transit  is  usually  plain,   but  it  is 
often  convenient  to  have  a  clamp  and  tangent  movement  to  tele- 


10        A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

scope,  a  vertical  circle,  a  level  on  telescope,  stadia  wires,  and  a 
gradienter;  the  solar  attachment  will  rarely  be  needed. 

20.  To  Adjust  the  Plate  Levels.— The  axis  of  the  instrument 
is  set  at  right  angles  to  the  plates  by  the  manufacturer,  so  that 
when  the  axis  is  made  vertical  the  plates  will  be  horizontal. 

In  making  adjustments  remember  that  a  complete  reversal 
always  doubles  any  existing  error. 

Place  the  bubble-tube  parallel  to  a  diagonal  pair  of  leveling- 
screws,  and  bring  Ihe  bubble  to  the  centre  of  its  run.  Revolve 
the  instrument  180°  on  the  vertical  axis,  and  the  level- tube  will 
be  parallel  to  the  same  pair  of  leveling-screws  as  before,  but 
reversed.  If  the  bubble  has  moved  from  its  central  position 
bring  it  half-\v&y  back  by  means  of  the  capstan-headed  screws  at 
the  ends  of  the  tube.  Relevel  and  repeat  until  the  bubble  remains 
at  the  centre  after  reversal.  Do  the  same  for  the  other  bubble. 
Both  bubbles  should  remain  at  the  centres  of  their  tubes  during  a 
complete  reversal. 

21.  Parallax  is  an  apparent  movement  of  the  cross- wires  with 
respect  to  the  object  sighted  when  the  eye  is  moved  from  side  to 
side  of  the  eyepiece,  and  shows  that  the  image  does  not  fall  in  the 
plane  of  the  cross-wires.     In  precise  measurements  it  should  be 
removed  before  making  an  observation  with  the  telescope.     To  do 
this,  first  bring  the  cross-wires  clearly  into  view  when  the  object- 
glass  is  turned  towards  the  sky,  then,  when  sighting  an  object, 
note  if  there  is  any  relative  movement  of  cross-wires  and  image 
when  the  eye  is  moved  from  side  to  side  at  the  eyepiece  ;  if  there 
is,  refocus  the  object-glass  until  this  movement  disappears. 

22.  To  Adjust  the  Line  of  Collimaiion  is  to  make  the  line 
joining  the  intersection  of  cross-wires  and  optical  center  of  objec- 
tive describe  a  plane  perpendicular  to  the  horizontal  axis  of  instru- 
ment. 

FIRST  METHOD. — Level  the  instrument  and  clamp  the  move- 
ments on  vertical  axis.  Sight  some  well-defined  object  distant 
about  the  length  of  an  average  sight,  and  in  the  same  horizontal 
plane  as  telescope.  Reverse  the  telescope  on  its  horizontal  axis, 
and  fix  a  point  about  as  far  from  instrument  as  first  point,  and  in 
the  same  horizontal  plane.  Revolve  the  instrument  on  its  vertical 
axis  and  sight  the  first  point;  then  reverse  the  telescope  and  note 
if  line  of  sight  cuts  the  second  point.  If  not,  loosen  the  capstan- 
headed  screws  holding  cross- wire  ring  and  move  the  vertical  wire 


PRELIMINARY    SURVEYS.  11 

over  one  fourth  the  apparent  error — since  tliere  were  two  reversals 
— remembering  that  the  image  of  the  cross-wires  is  inverted,  while 
that  of  the  object  appears  in  its  true  position.  Test  by  repetition. 

SECOND  METHOD.— If  the  limb  graduations  can  be  relied  on 
they  may  be  used  in  adjusting  the  vertical  wire.  With  the  instru- 
ment level  sight  a  well-defined  point,  then  revolve  180°  by  vernier- 
plate,  reading  both  verniers;  reverse  telescope,  and  note  if  line  of 
sight  cuts  the  point.  If  not,  correct  one  half  the  apparent  error  by 
moving  diaphragm  ;  then  test  by  repetition. 

The  manufacturers  adjust  the  object-glass  slide  so  that  the  ob- 
jective travels  in  the  telescope  axis,  and  this  adjustment  is  not 
liable  to  serious  derangement.  It  is  well,  however,  to  sometimes 
test  by  adjusting  the  line  of  collimation  for  both  near  and  distant 
objects.  If  not  correct  for  both,  move  the  ring  which  guides  the 
rear  end  of  object-glass  slide  until  the  adjustment  is  correct  for 
both  positions. 

Next  make  the  vertical  wire  vertical  by  noting  if  it  coincides 
throughout  its  length  with  a  plumb-line,  or  by  observing  if  it  de- 
viates from  a  point,  on  which  the  intersection  has  been  fixed,  when 
the  telescope  is  elevated  or  depressed.  Any  error  is  corrected  by 
turning  the  ring  after  slightly  loosening  the  screws  holding  it. 

The  horizontal  wire  should  also  be  adjusted  so  that  the  inter- 
section of  the  cross-wires  will  be  in  the  axis  of  the  telescope  ;  if 
the  transit  is  to  be  used  as  a  leveling  instrument  this  adjustment 
is  essential. 

Drive  a  stake  close  to  the  instrument,  and  with  the  telescope 
clamped  as  nearly  horizontal  as  can  be  conveniently  done  read  a 
rod  held  on  top  of  the  stake  ;  about  800  feet  distant,  and  in  line 
with  first  stake  and  instrument,  drive  a  second  stake  and  read  the 
rod  on  it.  Revolve  180°  on  vertical  axis,  reverse  the  telescope  and 
bring  the  horizontal  wire  to  the  former  reading  when  the  rod  is 
held  on  first  stake  ;  if  the  reading  on  the  second  stake  is  not  the 
same  as  before,  correct  one  half  the  apparent  error  by  moving  the 
cross-wire  ring.  Repeat  as  a  test.  The  vertical  wi,re  should  again 
lie  tested  lest  the  movement  of  the  ring  may  have  caused  it  to 
change. 

23.  To  Adjust  the  Standards  is  to  make  the  plane  described 
by  the  line  of  collimation  vertical.  Set  up  the  transit  about  as  far 
in  front  of  some  high  building,  or  other  tall  object,  as  the  highest 
point  that  can  be  sighted  is  above  the  base.  Level  the  instrument 
and  fix  the  intersection  of  the  cross-wires  on  the  highest  point  that 


12        A   FIELD-MANUAL   FOR  RAILROAD   ENGINEERS. 

can  be  easily  sighted.  Depress  the  telescope  and  fix  a  point  near 
the  base  of  the  building  at  about  the  height  of  the  telescope.  Un- 
clamp  and  revolve  on  the  vertical  axis  until  the  telescope  reversed 
cuts  the  lower  point.  Clamp  the  plates  and  raise  the  telescope 
until  the  cross-wires  are  at  the  height  of  the  upper  point.  If  they 
cut  it  the  standards  are  in  adjustment.  If  they  do  not,  bring 
them  half-way  back  by  means  of  the  adjustable  screws  at  the  top 
of  one  of  the  standards.  Repeat  as  a  test. 

24.  To  Adjust  the  Level  on  Telescope  is  to  make  the  bubble 
stand  at  the  center  of  its  run  when  the  line  of  sight  is  horizontal. 
Bring  the  telescope  as  nearly  horizontal  as  maybe  convenient,  and 
take  readings  on  the  tops  of  two  pegs  in  the  same  vertical  plane 
with,  and  equidistant  from,  the  instrument — say  300  feet.  The 
difference  of  readings  will  equal  the  difference  of  elevation  of  the 
pegs;  this  difference  may  be  obtained  with  the  wye-level  if  pre- 
ferred. 

Move  the  instrument  to  a  point  beyond  one  of  the  pegs  and  in 
line  with  both.  Set  up  as  close  to  nearer  peg  as  convenient,  but 
not  so  close  that  the  rod  cannot  be  easily  read.  Bring  the  tele- 
scope as  nearly  horizontal  as  possible,  and  read  on  both  pegs.  If 
the  difference  of  readings  equals  their  difference  of  elevation  the 
line  of  sight  is  horizontal,  and  the  bubble  may  be  brought  to  the 
center  by  means  of  the  adjustable  screws  attaching  the  level-tube 
to  the  telescope.  If  this  is  not  the  case,  we  must  set  the  telescope 
so  the  reading  on  second  peg  equals  the  reading  on  first  peg  plus 
the  difference  of  elevation  ;  then  read  again  on  first  peg  and  pro- 
ceed as  before  until  the  condition  is  satisfied.  Or  we  may  proceed 
as  follows : 

In  Fig.  1  let  the  transit  be  at  0,  and  A  and  B  be  the  pegs.  AC 
is  a  horizontal  through  A,  so  that  CB  is  the  difference  of  elevation 


of  A  and  B.     Suppose  line  of  sight  to  cut  the  rods  at  7?  and  D, 
we  must  find  DO  so  that  the  target  may  be  set  at  the  proper  read- 


PRELIMINARY    SURVEYS.  13 

ing  to  make  the  line  of  sight  horizontal.     Let  OF=  a,  FG  =  b, 
EA  =  r,  DB  =  r,  CD  —  k.     Draw  DH  parallel  to  CA  and  OG; 
then  ER=  r  -\-k-r'. 
From  similar  triangles 


Set  the  target  at  a  reading  GB  =  GD  +  ?•',  sight  to  G,  and  the 
line  of  sight  will  be  horizontal.  Bring  the  bubble  to  the  center  of 
its  run  while  the  telescope  is  in  this  position,  and  the  adjust- 
ment is  complete. 

If  desired,  a  correction  for  the  curvature  of  the  earth  and  re- 
fraction may  be  introduced,  but  for  short  sights  this  is  a  useless 
refinement. 

25.  The  Vernier  is  an  auxiliary  scale  for  measuring  smaller 
divisions   than   those   graduated   on   the   limb.     There   are   two 
classes,  the  direct-reading  and  the  retrograde,  according  as  the 
fractional  parts  of  limb  readings  are  taken  on  that  side  of  the 
zero  of  vernier  scale  towards  which  the  vernier  has  moved  with 
respect  to  the  limb,  or  the  reverse.     On  the  direct  vernier  a  cer- 
tain number  of  divisions  on  the  vernier  equals  the  same  number 
of  divisions  on  the  limb,  less  one  ;  on  the  retrograde  there  is  one 
more  division  on  limb  than  on  vernier  when  the  same  space  is 
covered  by  both. 

26.  The  Least  Count  of  a  vernier  is  the  smallest  subdivision  of 
limb  graduation  that  can  be  read  by  it,  and  equals  the  difference 
of  one  space  on  limb  and  one  on  vernier. 

Let  I  =  value  of  one  space  on  limb  ; 

v  =  value  of  one  space  on  vernier  ; 

n  =  number  of  spaces  on  vernier. 
Then  for  the  direct  vernier 

nv  =  (n  —  1)1  ; 
from  which  we  get  the  least  count, 


For  the  retrograde  vernier 

nv  =  (n  -f-  IX, 


14        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

from  which  the  least  count  is 

,-l=L, 

n 

the  same  result  as  found  for  the  direct  vernier. 

So,  to  find  the  least  count :  Divide  the  value  of  one  limb  space  "by 
the  number  of  spaces  on  the  vernier. 

For  example  :  If  the  limb  of  a  transit  is  divided  to  half-degrees 
and  the  number  of  spaces  on  the  vernier  is  30,  the  least  count 
will  be  ^  divided  by  30,  or  ^  of  a  degree — that  is,  1  minute. 

27.  To  Read  a  Vernier,  take  the  number  of  the  last  division  on 
limb  back  of  the  vernier  zero,  then  look  along  the  vernier  until  a 
line  is  found  to  coincide  with  a  line  on  the  limb  ;  add  the  number 
of  this  vernier  line,  multiplied  by  the  least  count,  to  the  scale 
reading,  and  the  result  will  be  the  required  reading. 

C,   Accessories. 

(1°)  The  Oradienter. 

28.  The   Gradienter  consists     of   a  tangent-screw  having   a 
micrometer-head,  attached  to  one  of  the  standards  of  the  transit 
and  capable  of  being  clamped  to  the  horizontal  axis  of  the  tele- 
scope.    It  is  used — as  its  name  indicates — in  running  grades,  and 
it  accurately  measures  a  small  vertical  angle  in  terms  of  its  tan- 
gent.    The  screw  is  so  cut  that  one  revolution  moves  the  tele- 
scope through  an  angle  whose  tangent  at  one  hundred  feet  from 
the  instrument  has  a  certain  value,  usually  one  foot.     The  grad- 
uated head  is  divided  into  100  parts,  so  that  one  division  corre- 
sponds to  0.01  ft.  at  100  feet  from  instrument. 

To  run  a  given  gradient,  bring  the  telescope  level  and  read  the 
micrometer-head  of  screw;  then  turn  the  screw  as  many  divisions 
as  there  are  hundredths  of  a  foot  rise  or  fall  in  100  feet,  and  with 
a  target  set  at  the  height  of  the  horizontal  axis,  points  on  the 
surface,  corresponding  to  the  given  grade  can  be  found. 

For  example  :  To  run  a  0.75  per  cent  grade,  move  the  microm- 
eter milled  head  75  graduations  from  the  horizontal. 

When  used  as  a  Telemeter,  we  may  either  measure  the  space 
on  the  rod  moved  over  by  the  line  of  sight  for  a  given  number  of 
revolutions  of  the  screw,  or  we  may  note  the  number  of  revolu- 
tions required  to  move  the  line  of  sight  over  a  certain  space  on 
rod.  The  second  method  is  the  more  accurate,  particularly  for 
long  sights. 


PRELIMINARY    SURVEYS. 


15 


(2°)  The  Stadia,  or  Telemeter. 

29.  The  Stadia  is  an  instrument  for  determining  the  distance 
of  a  point  from  the  observer  by  noting  the  space  intercepted  on  a 
rod  by  a  given  visual  angle,  as  determined  by  two  auxiliary  wires 
parallel  to,  and  equidistant  from,  the  horizontal  wire  of  the  transit 
telescope.     When  used  with  an  ordinary  leveling-rod  the  wires 
should  be  adjustable ;  if  they  are  fixed  (which  for  some  reasons 
is   preferable),   the   rod   must   be   graduated  to   correspond.     In 
addition  to  the  distance  of  a  point  from  the  instrument,  the  differ- 
ence of  elevation  is  determined  by  observing  the  angle  made  by 
line  of  sight  with  the  horizontal  when  the  middle  horizontal  wire 
cuts  a  point  on  the  rod  as  high  above  the  ground  as  is  the  centre 
of  the  telescope. 

The  horizontal  position  of  the  point  is  determined  from  its 
magnetic  bearing,  or  the  azimuth  of  line  of  sight  with  reference 
to  some  fixed  line,  usually  the  north-south  line. 

30.  Line  of  Sight  Horizontal.— In  Fig.  2  let  a  and  b  be  the 

stadia  wires,  AB  the  intercept  on  the  rod.     The  secondary  axes 

A 


Fro.  2. 


aA  and  bB  pass   through   the   optical   center  0.     Let   h  =  ab, 
r  =  AB,  d  =  distance  of  cross-wires  from  objective,  D  ~  distance 
of  rod  from  objective. 
From  similar  triangles, 


From  optics, 

l  +  i  =  L 

d  ^  D      f 

in  which/  is  the  focal  length  of  objective. 
Eliminating  d  from  these  two  equations, 


D  -. 


16        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


Let  c  be  the  mean  distance  of  objective  from  center  of  instru- 
ment. Adding  this  to  D  gives,  for  the  distance  of  the  rod  from 
the  center  of  the  instrument, 


-  may  be  made  constant,  when  (2)  becomes 


(2) 


(2') 


31.  Line  of  Sight  Inclined. — When  the  line  of  sight  is  not 
level  it  is  difficult  to  hold  the  rod  perpendicular  thereto  ;  hence 
the  rod  is  held  vertical,  the  angle  of  inclination  measured,  and  a 
correction  applied.  In  Fig.  3 

Q. 
F, 


FIG.  3. 

let  r  =  CD  be  the  reading  on  rod  held  vertical ; 

r'  =  FE,  the  reading  perpendicular  to  line  of  sight  ; 
H  =  AG,  the  horizontal  distance  from  A  to  B  ; 
V=  BO,  the  difference  of  elevation  between  A  and  B ; 
n  =  BAG,  the  angle  of  inclination  of  line  of  sight. 

Assume  angles  AFB  and  AEB  =  90°,  from  which  they  rarely 
differ  more  than  15'  to  17'.     Then,  since  FBC  =  n, 

r'  —  r  cos  n. 


PRELIMINARY    SURVEYS.  17 

By  (2').  AB  =  a  +  kr'. 

Hence  AB  =  a  -f  kr  cos  n. 

From  triangle  ABG 

H  =  AB  cos  n 

.-.  H  =  a  cos  w  -f-  kr  cos2  ?i (3) 

V=  AB  sin  T&; 

.*.  V  =  a  sin  n  -j-  A;?'  sin  n  cos  n. 
But  2  sin  n  cos  TZ.  =  sin  2n. 
Hence  V  =  a  sin  n  -j-  ^Ar  sin  2tt (4) 

32.  The  Instrumental  Constant  a  [—  c  +  /  of  (2)]   may  be 
found  by  measuring   the  distance  from  center  of  instrument  to 
mean  position  of  objective,  which  equals  c  ;  then  focusing  on  a 
very  distant  object,  preferably  a  star,  and  measuring  from  center 
of  objective  to  plane  of  cross-wires,  which  equals/.     The  sum  of 
these  distances  is  a  in  formulas  (3)  and  (4). 

If  the  stadia  wires  are  fixed,  k  may  be  found  by  measuring  for- 
ward on  level  ground  the  distance  a  from  plumb-line,  and  from 
this  point  a  further  distance  b  ;  then  note  carefully  the  stadia 
reading  r  when  the  telescope  is  level.  Then,  remembering  (2)', 

a  -\-  b  =  a  -{-  kr. 

.'.  k  —  — ,  a  constant  ratio. 
r 

If  the  stadia  wires  are  adjustable,  we  may  so  adjust  k  that  any 
desired  reading  may  be  had  for  a  given  length  of  base.  A  con- 
venient value  of  k  is  100,  which  corresponds  to  an  intercept  of 
1  foot  on  the  rod  at  100  feet  from  a  point  a  feet  in  front  of  the 
instrument,  2  feet  at  200  feet  in  front,  etc. 

33.  A  Stadia  Table  based  on  formulas  (3)  and  (4)  is  published 
by  the  D.  Van  Nostrand  Company  in  Winslovv's  Stadia  Surveying, 
and  can  be  used  more  rapidly  than  the  formulas.     Johnson's  Re- 
duction Diagram,  by  John  Wiley  &  Sons,  gives  values  of  //and  V 
graphically.     Colby's   Slide-rule,   manufactured  by  Mahn  &  Co., 
St.  Louis,  gives  values  of  V  for  distances  in  feet,  yards,  or  meters 
to  tenth:  of  a  foot,  and  can  be  used  with  great  rapidity. 


18        A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


•    D.   Field-work. 

34.  Station  Numbers  should  begin  with  zero  for  the  initial 
stake,  and  are  marked  on  rear  side  of  stake,  from  the  top  down- 
ward, the  number  of  the  preliminary,  A,  B,  G,  stc.,  being  marked 
on  the  forward  side.     The  marking  should  be  with  kiel,  or  crayon 
that  will  withstand  the  action  of  sun  and  rain.     Stakes  may  be 
set  every  hundred  feet  or  only  at  even  stations,  as  preferred. 

35.  Hubs,  or  Plugs,  are  transit  turning-points,  and  are  short, 
flat-topped  stakes  driven  into  the  ground  flush  with  the  surface. 
The  flag  is  held  on  the  top  and  carefully  aligned,  the  position  of 
the  point  being  marked  by  a  tack.      A  special  tack  with  concave 
head  offers  a  foothold  for  point  of  flag  when  used  in  backsight- 
ing.*     About  10  inches  to  the  left  of  and  with  numbered  side 
facing  the  hub  is  driven  a  guard-stake  to  mark  its  position. 

36.  Reference-points  are  two  or  more  hubs,  with  guard-stakes, 
in  each   of   two  lines  making  a  good  intersection  angle  at  the 
point  whose  position  they  serve  to  locate.     They  should  be  driven 
beyond  reach  of  disturbance,   and  are  used  in  replacing  a  dis- 
located hub. 

These  need  rarely  be  used  on  preliminary. 

37.  Alignment. — It  is  not  intended  that  the  preliminary  and 
location  lines  occupy  exactly  the  same  position  ;  hence  consider- 
able  latitude   is   allowable   in   the   size   and   number  of  angles 
turned,  care  being  taken,  however,  that  the  maximum  curvature 
need  not  be  exceeded  on  location.     Large  trees  and  other  obstruc- 
tions may  be  avoided  by  turning  a  small  angle  until  the  obstacle 
has  been  passed,  then  making  a  deflection  in  the  opposite  sense. 
Bearings  of  tangents  are  taken  with  the  needle,  to  serve  as  a 
check  on  the  angle  read  on  the  plates. 

In  easy  country  not  requiring  a  topographic  party  large  angles 
should  not  be  turned,  a  succession  of  small  ones  with  short  inter- 
vening tangents  being  substituted  in  order  to  make  the  prelimi- 
nary profile  approximate  more  closely  to  the  location  profile. 
These  short  tangents  may  conveniently  be  the  long  chords  of  the 
curve  that  is  to  follow. 

*  Such  a  tack  is  manufactured  by  the  A.  S.  Aloe  Co.,  St.  Louis. 


PRELIMINARY    SURVEYS. 


19 


38.  The  Transit  Notes  may  be  kept  in  the  form  below,  which 
shows  both  pages  of  the  note  book.  Tfee  notes  run  from  the 
bottom  up,  the  right-hand  page  being  reserved  for  sketches  ;  the 
red  line  up  the  middle  of  the  page  represents  the  transit  line, 
whether  straight  or  broken,  to  which  the  sketches  must  be 
adjusted. 


Sta. 

Angle. 

Calculated 
Course. 

Magnetic 
Course. 

Remarks  and  Sketches. 

68 
670 
66 
65 
64 
630 

20°  0'  L. 
6°2'R. 

N.  1°  48'  W. 

N.  18°  12'  E. 

N.  1°  45'  W. 

N.  18°  15'  E. 

( 

c 

) 

61 

39.  Stadia  Methods  for  Preliminary  Surveys. — Preliminary 
lines  are  usually  run  with  the  transit,  but  the  compass  will 
answer  nearly  as  well  in  most  cases,  besides  admitting  of  more 
rapid  work.  The  transit  and  stadia  method  might  well  be  em- 
ployed, and  would  effect  considerable  saving  in  the  cost  of  pre- 
liminary surveys.  For  some  reason  railroad  engineers  have  not 
regarded  it  with  favor,  though  it  is  extensively  employed  in 
topographic  surveying  where  the  map  is  to  be  used  for  work  that 
is  often  more  precise  than  needed  for  railroad  preliminaries. 

Particularly  is  this  method  applicable  to  exploration  lines. 
With  the  transit  and  stadia  the  entire  surveying  corps  need  not 
exceed  five  or  six  men,  the  instrument  man  acting  as  transitman, 
leveler,  and  topographer  all  in  one.  The  only  objection  would 
seem  to  be  in  the  amount  of  reduction  the  notes  would  need ; 
however,  with  tables  and  slide-rule  (see  33)  this  work  may  be 
very  rapidly  done.  For  vertical  angles  of  less  than  one  degree 
the  horizontal  reduction  can  be  neglected,  and  with  side  readings 
for  topography  the  angle  may  be  5  or  10  degrees  without  necessi- 
tating the  correction.  Vertical  heights  are  found  by  the  slide- 
rule  or  by  charts. 

This  method  would  really  necessitate  the  making  of  a  topo- 
graphic map  along  a  narrow  strip  of  country,  from  which  the 
profile  could  readily  be  taken.  With  a  skilled  observer  and  two 
to  four  rodmen  the  progress  may  be  more  rapid,  and  fully  as 
good  for  the  purpose  intended  as  the  more  expensive  method 
usually  employed, 


20        A   FIELD-MANUAL   FOR    RAILROAD    p:\OINEKRS. 

The  transit  need  only  be  set  at  alternate  stations  (which  may  be 
any  length  within  the  reading  limits  of  the  wires),  the  bearings  to 
other  stations  and  points  off  the  line  being  taken  with  the  needle. 
The  horizontal  angle  should  also  be  read  on  the  plates  for  points 
on  stadia  line,  as  a  check  on  the  bearings. 

E.  Obstacles  in  Tangent. 

40.  Obstructions  to  vision  and  measurement  in  tangent  may  be 
avoided  in  a  number  of  ways,  a  few  of  which  are  given  in  the 
following  problems.     Other  methods  of  avoiding  them  will  sug- 
gest themselves  in  special  cases. 

The  same  devices  may  be  used  on  location,  but  it  is  more  im- 
portant to  maintain  a  clear  sightvvay  then  ;  so0  when  possible,  we 
should  remove  the  obstruction. 

41.  To  Pass   an  Obstacle  by  Means  of  Parallel  Lines. — In 
Fig.  4,  0  is  the  obstruction,  AB  the  obstructed  line.     At  B  set 


FIG.  4. 

transit  ;  turn  90°  and  measure  BF  long  enough  to  clear  obstruc- 
tion. Set  transit  at  F,  make  BFG  —  90°,  and  measure  FG. 
Move  to  G  and  backsight  to  F,  making  FGC  =  90°.  Measure 
GC  —  FB,  and  move  to  C,  where  the  angle  GCD  is  made  equal 
to  90°.  CD  is  the  desired  line,  and  EC  =  FG. 

Otherwise,  at  A  and  B  erect  perpendiculars;  take  BF=AE; 
produce  EF,  and  at  G  and  II,  beyond  0,  erect  perpendiculars  mak- 
ing GG  =  HD  —  FB.  CD  will  be  the  desired  line,  and  BC  =  FG. 

42.  To  Pass  an  Obstacle  by  Angular  Deflections. 

GENERAL  CASE.     Angle  anything  less  than  90°. 

At  B  (Fig.  5)  on  the  obstructed  line  deflect  an  angle  a  to  one 
side  and  measure  BC,  taking  C  so  that  after  deflecting  2a  to  the 
other  side  CD  will  clear  the  obstruction.  Make  CD  =  BC  and 
deflect  an  angle  a  to  the  same  side  as  at  B;  DE  will  lie  in  AB 
produced.  Draw  CH  perpendicular  to  BD;  then 

cos  a (5) 


PRELIMINARY   SURVEYS. 


21 


&__ 


FIG.  5. 

EXAMPLE.— Suppose  a  =  14°  10',  BC=CD  =  520  ft. 
BD  =  2  X  520  X  0.96959  =  1008.37  feet. 
SPECIAL  CASE.     Angle  60  degrees. 

In  this  case  the  triangle  BDF(Fig.  6)  is  equilateral  and  BF  = 
BD  =  DF. 

Should  it  be  inconvenient  to  run  to  D  we  may  stop  at  C,  having 
measured  BC.     At  C  deflect  60°  and  measure  CE;  at  E  again  de- 


FIG.  6. 

fleet  60°  and  make  EF=  BC.     At  F  a  final  deflection  of  60°  in  the 
opposite  sense  will  put  the  telescope  in  the  desired  line,  FG,  and 
BF=BC+CE.     ......     (5a) 

43.  To  Pass  an  Obstruction,  such  as  a  River,  when  the  Pre- 
ceding Methods  are  Inapplicable. 
FIRST  CASE.     Point  beyond  obstruction 
In  Fig.  7  let  BC  be  required. 

.i.     c       , 


FIG.  7. 

At  7?  erect  and  measure  the  perpendicular  BD  ;  set  instrument 
at  D  and  measure  angle  BDC  =  a  ;  then 

BC=BDta,na (6) 


22        A    FIELD-MANUAL   FOR    RAILROAD   ENGINEERS. 

Or,  if  a  trigonometric  table  is  not  at  hand,  make  CDE  —  90°  and 
fix  the  point  E  where  DE  intersects  AB  ;  measuring  EB  there 
results,  from  similar  triangles, 

CB_BD 
BD  ~  EB' 

B2)'2 
whence  CB  =-f^-  ........     (6«) 

hih  v 

Otherwise,  if  a  right  angle  at  B  is   not  convenient,   measure 
angles  CBD  =  b,  BDC  =  a,  and  side  BD.    Then  c  =  180°-  (a+6). 
From  triangle  BDC, 

BC  =  BD^  ......     (66) 

sin  c 

EXAMPLE.—  a  =  56°,  b  =  70°,  £Z>  -  400  feet. 

sin  5fi° 

By  (66),  J5C  =  400  ^T    ^  =  409.8  feet. 

sin  54 


SECOND  CASE.     Poiw^  beyond  obstruction  invisible. 
At  B  (Fig.  8)  measure  angle  &  and   line  BE;  move  to  E  and 
measure  angle  y,  and  set  hubs  on  line  EG  so  the  line  BC  will  pass 


FIG.  8. 

between  them.     Angle  z  =  ECB  =  180  -  (b  -f  y\    Then  from  tri- 
angle BEG 

sin  z  ' 

Produce  EB  to  D,  where  DC  will  be  sure  to  clear  obstruction ; 
measure  BD. 

From  triangle  BDC, 

tan  \(a  -  x)  _  BC  -  BD 


But  «  +  «  =  &,  hence 

tan  \(a  -  *)  =  •  tan  tf .        ...     (8) 


PRELIMINARY    SURVEYS.  23 

The  sum  and  difference  of  a  and  x  are  now  known,  so  both  may 
be  readily  found. 

At  D  set  off  the  angle  a  with  the  transit,  and  have  the  chaiumen 
stretch  a  cord  between  the  hubs  set  on  line  EC  at  C.  Now  signal 
the  flagman  to  move  his  rod  along  this  cord  until  the  vertical  wire 
cuts  it  at  C.  Set  a  hub  here  and  place  the  transit  over  it.  Sight 
to  D  or  E,  reverse  telescope  and  deflect  into  CH. 

ARTICLE  4. — THE  LEVEL  PARTY. 

44.  The  Level  Party  consists  generally  of  two  members,  the 
leveler  and  a  rodman ;  sometimes  an  axeman  is  added  to  keep  the 
rod  man  supplied  with  pegs  for  turning-points  and  in  clearing  the 
line  of  sight  for  the  level.     As  the  party  follows  the  transit  little 
or  no  clearing  will. be  needed.     The  instruments  used  are  a  level, 
a  rod,  and  a  hand-axe  or  hatchet. 

45.  The  Leveler  makes  all   necessary  observations  with  his 
instrument,  keeping  a  neat,  accurate  record  of  readings  and  ele- 
vations ;  also  the  positions  and  elevations  of  benches  and  turning- 
points.     He  should  work  out  elevations  of  stations  while  the  rod- 
man is  going  from  one  station  to  the  next ;  he  must  see  that  the 
rodman  gives  him  readings  at  points  where  the  longitudinal  slope 
changes  suddenly,  recording  the  plus.     He  must  plot  his  profile 
at  night,  or  at  such  times  as  the  chief  of  party  is  likely  to  need  it. 
The  rodman 's  readings  at  turning-points  should  be  checked. 

46.  The  Rodman  holds  his  rod  at  each  station,  calling  out  the 
number.    If  stakes  are  set  only  at  even  stations,  he  must  hold  his 
rod  midway  between  stakes,  the  point  being  found  by  pacing  the 
distance.     Target-readings  need  only  be  taken  at  turning-points 
and  benches,  and  the  rodman  should  keep  a  record  of  these  in 
his  "peg-book,"  checking  the  calculations  of  leveler  for  heights 
of  instrument  and  elevations  of  turning-points.     At  any  marked 
surface  change  he  will  hold  his  rod,  calling  out  the  plus  to  leveler. 
He  must  assist  the  leveler  in  plotting  up  the  notes. 

A.  Adjustments  of  the  Level. 

47.  To  Adjust  the  Line  of  Collimation  is  to  bring  the  inter- 
section of  the  cross-wires  into  the  optical  axis  of  the  telescope. 

Set  up  and  level  the  instrument,  then  bring  the  vertical  wire 
into  coincidence  with  a  plumb  line  or  vertical  edge  of  a  building, 


24        A   FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

at  the  mean  length  of  sight,  and  note  if  the  vertical  wire  is  truly 
parallel  thereto.  If  it  is  not,  loosen  the  capstan-headed  screws 
holding  cross-wire  ring  and  turn  slightly  so  that  the  wire  is 
parallel  to  the  vertical  line. 

Loosen  the  wye-clips  and  bring  the  vertical  wire  into  coin- 
cidence with  the  line  and  clamp  the  instrument.  Rotate  the 
telescope  in  the  wyes  180°  and  note  if  the  wire  coincides  with  the 
line.  If  not,  correct  one  half  the  error  by  loosening  one  and 
tightening  the  opposite,  of  the  capstan -headed  screws  that  hold 
the  cross-wire  ring  in  place,  remembering  that  the  image  of 
the  cross  wires  is  inverted  by  the  eyepiece. 

Turn  the  telescope  until  the  horizontal  wire  is  parallel  to  the 
plumb-line  or  edge  of  building,  and  make  the  same  test  and 
correction.  Repeat  for  both  wires.  The  horizontal  wire  is  the 
one  on  which  the  accuracy  of  leveling  depends,  but  it  is  wise  to 
have  both  adjusted.  Their  intersection  should  remain  on  a  point 
during  a  complete  rotation  of  the  telescope  in  the  wyes. 

48.  To  Adjust  the  Level-bubble  is  to  bring  the  axis  of  tho 
level-tube  into  the  same  vertical  plane  with  the  line  of  collimation, 
and  to  make  the  bubble  stand  at  the  center  when  the  line  of  sight 
is  horizontal. 

Since  the  axis  of  the  telescope  coincides  with  the  line  joining 
the  center  of  the  wye-rings  (which  requires  these  to  be  of  the 
same  size),  it  is  sufficient  to  make  the  axis  of  the  bubble  parallel 
to  this  line. 

(a)  With  the  telescope  over   one  diagonal   pair  of  leveling- 
screws  and  the  clips  loosened,  bring  the  bubble  to  the  center  of 
its  run  ;  then  turn  the  telescope,  in  the  wyes,  a  little  to  either  side 
of  the  vertical  plane  through  the  telescope  and  note  if  the  bubble 
remains  at  the  center.     If  not,  correct  the  error  by  means  of  the 
screw  at  end  of  the  level-tube  case  arranged  for  lateral  movement. 
Repeat  until  the  tube  may  be  rotated  half  an  inch  or  more  to 
either  side  of  vertical  without  movement  of  the  bubble.     This 
adjustment  is  made  merely  to  prevent  error  from  failure  to  set 
level-tube  vertically  beneath  telescope. 

(b)  With  the  wye-clips  opened  well  out,  again  bring  the  bubble 
to  the  center  of  its  run  ;  remove  the  telescope  from  wyes  and 
turn  it  end  for  end,  then  carefully  replace  it  in  the  wyes.    Should 
the  bubble  fail  to  remain  at  the  center,  bring  it  half-way  back  by 
raising  the  lower  or  depressing  the  higher  end  of  tube   at    ilic 
points  of  attachment  to  telescope.     Relevel  and  repeat  as  a  lest. 


PRELIMINARY   SURVEYS. 


49.  To  Adjust  the  Wyes  is  to  make  the  axis  of  the  telescope 
perpendicular  to  the  vertical  axis.      With   the  wye-clips  closed 
place  the  telescope  over  one  pair  of  leveling-screws  and  bring  the 
bubble  to  the  center  of  its  run  ;  then  turn  the  telescope  half-way 
round  on  its  vertical  axis,  so  that  its  ends  have  changed  places. 
If  there  is  any  error,  correct  by  bringing  the  bubble  half -way  back 
to  center  by  means  of  the  screws  connecting  wyes  with  level-bar. 
Repeat  until  the  bubble  remains  in  the  center  during  a  complete 
revolution. 

B.    Theory  of  Leveling. 

50.  When  the  level  has  been  adjusted  the  line  of  collimation 
will  describe  a  plane  parallel  to  the  horizontal  plane  tangent  to 
the  earth's  surface  at  the  point  where  the  instrument  is  placed. 
A  level  surface,  such  as  the  surface  of  still  water,  will  coincide 
with  this  plane  only  at  the  point  of  taugeucy,  and  will  depart 
farther  and  farther  therefrom  as  the  point    considered  recedes 
from  the  instrument.     For  short  sights  this  difference  may  be 
neglected  in  railroad  work,  as  will  presently  be  shown,  but  for 
long  sights  a  correction  must  be  applied. 

The  effect  of  curvature  is  to  make  objects  appear  lower  than 
they  really  are,  while  the  refraction  of  a  beam  of  light,  due  to 
the  greater  density  of  the  layers  of  air  nearest  the  earth's  surface, 
has  a  contrary  effect.  Experience  shows  the  average  error  due 
to  refraction  to  be  about  one  seventh  of  that  due  to  curvature. 

51.  The  Error  due  to  Curvature  at  any  point  is  the  deviation 

of  a  tangent  line  from  true  level,  as  -j^ t  p 

the  point  recedes  from  the  point  of 

tangency. 

Let  0  be  the  center  of  the  earth,  T 
the  point  of  taugeucy,  and  j^the  point 
where  the  error  due  to  curvature  is 
desired.  Let  the  notation  be  as  shown 
in  Fig.  9.  From  the  right  triangle 
OTP,  we  have 

(R  +  c)2  =  IP  +  t*. 

From  which 

t* 

=  2M  f  e  ' 

Now,  since  c  is  always  very  small  compared  with  27?,  the 
quotient  resulting  from  the  division  of  t-  by  27?  will  not  differ 


FIG.  9. 


26        A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

sensibly  from  that  obtained  by  dividing  by  2J?  +  c.     Therefore 
we  write 


c  = 


(9) 


For  t  =  1  mile,   H  =  3963  miles,    c  =  about  8  inches.      Hence 
for  any  other  distance  in  miles  we  have,  for  c, 

c  =  S  X  t*  inches (9«) 

The    correction   for  refraction  is  about   -^c,  hence  we  have, 

from  (9), 

n  1         6         3  <2 

C^c--c  =  -c=-- 

or,  closely  enough, 

G  =  .85c (10) 

EXAMPLE.— What  is  the  correction  for  a  half-mile  sight? 
For  one  eighth  of  a  mile? 

By  (9«),        c  =  8  X  (£)9  =  2"  for  first  case, 
and  c  =  S  X  (I)2  =  0".125  for  second  case. 

By  (10)  the  final  correction  is 

c  =  0.85  X  2  =  1".7  for  first  case, 

c  =  0.85  X  0.125  =  0.106"  for  second  case. 

52.  The  Difference  of  Elevation  between  two  points  not  so 
far  apart  but  that  a  rod  may  be  read  on  each  from  some  inter- 
mediate point  may  be  readily  found  from  these  rod-readings. 

In  Fig.  10  let  the  instrument  be  at  1,  A  and  B  the  points 
whose  difference  of  elevation  is  desired.  Let  r  =  AD,  r'  =  BC. 
Since  the  line  of  sight,  DO,  is  horizontal,  the  difference  of 

C 


FIG.  10. 

elevation  will  evidently  be  if  —  r.  When  the  distance  from 
/  to  A  equals  that  from  I  to  £  the  errors  due  to  curvature 
evidently  balance. 


PRELIMINARY   SURVEYS.  £7 

When  the  points  are  so  situated  that  the  rod  cannot  be  read 
on  both  from  one   intermediate   position  of  the  instrument,  an 


FIG.  11. 

auxiliary  point  or  points  must  be  used  and  readings  taken  on 
these  points  in  pairs.  Thus  in  Fig.  11  suppose  the  difference 
of  elevation  of  A  and  B  required  : 

With  the  instrument  "at  /  read  on  A  and  some  intermediate 
point  E.  Considering  the  backsights  as  plus  and  foresights  as 
minus,  the  difference  of  elevation  of  A  and  E  is  AD  —  FE. 

Again,  with  the  instrument  at  /'  the  difference  of  elevation  of  E 
and  B  is  OE—  CB.  The  sum  of  these  differences  equals  the  dif- 
ference of  elevation  of  A  and  B,  and  may  be  written  (AD  -\-  GE) 
—  (EF  -{-  CB),  or,  in  general,  the  sum  of  the  backsights  less  the  sum 
of  the  foresights  equals  the  difference  of  elevation. 

C.  Field-work. 

53.  A  Datum  is  a  level  surface  so  taken  that  it  shall  lie  below 
the  lowest  point  likely  to  be  reached  by  the  profile,  to  which  the 
surface  elevations  are  referred.     It   is   often   spoken   of  as  the 
datum-line  or  datum-plane,  and  is  the  zero  of  elevations. 

54.  A  Bench-mark  is  a  permanent  mark,  such  as  a  copper  or 
other  bolt  let  into  the  top  of  a  solidly  fixed  stone,  whose  height 
above  the  datum  is  known;  it  may  be  simply  a  mark  on  a  stone, 
or  a  tack  driven  into  the  projecting  root  of  a  tree,  upon  which 
the  rod  may  be  read.     In  any  case  it  must  be  so  situated  that  it 
cannot  change  its  elevation  nor  is  likely  to  be  disturbed  within 
the  time  for  which  it  is  intended  to  be  used  as  a  standard  of 
reference. 

The  elevation  should  be  marked  on  some  object  adjacent  to 
the  bench,  with  the  letters  B.  M.  indicating  the  nature  of  the 
point. 


28        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

55.  The  Field-work   consists  in  finding  the  elevation  of  a 
number  of  points  on  the  line  established  by  transit  party  suffi- 
cient to  give,  when  plotted,  a  fairly  correct  outline  of  the  surface 
as  seen  in  profile. 

A  bench-mark  is  taken  at  the  beginning  of  the  line,  and  its  dis- 
tance above  mean  sea-level  or  other  datum  is  known  or  assumed. 
The  level  is  set  with  one  pair  of  leveling  screws  in  the  line  to  be 
run  (in  order  that  any  change  in  the  position  of  the  bubble  may 
be  easily  corrected),  and  the  rod  is  read  on  the  bench.  This  read- 
ing plus  the  elevation  of  bench  gives  the  height  of  instrument 
(//.  /.)  above  the  datum. 

Readings  are  taken  at  every  hundred  feet  along  the  line,  or 
oftener  if  the  surface  changes  greatly,  until  a  point  is  reached 
beyond  which  it  is  desired  to  move  the  level.  A  peg  is  driven 
firmly  into  the  ground  and  the  rod  read  on  this ;  the  height  of 
instrument  less  the  rod  reading  will  give  its  elevation,  as  it  will 
for  the  intermediate  points.  This  point  is  a  temporary  bench 
and  is  called  a  turning-point.  It  should  be  marked  by  a  guard- 
stake  if  it  is  desired  to  use  it  again.  The  instrument  is  now  car- 
ried beyond  the  turning-point,  set  up,  and  the  whole  process 
repeated.  Benches  and  turning-points  should  be  read  to  hun- 
dredths  or  thousandths  of  a  foot,  intermediate  points  to  tenths. 
Turning-points  are  marked  0  or  T.  P.  in  the  notes,  and  their 
positions,  as  also  the  bench-marks,  noted  by  both  leveler  and 
rodman  in  their  note-books. 

56.  The  Level  Notes  may  be  kept  in  any  convenient  form 
that  is  easily  understood.     The  following  is  used  more  exten- 
sively, perhaps,  than  any  other: 


Sta. 

B.  S. 

H.I. 

F.  S. 

Elev. 

Remarks. 

B.M. 

5.613 

205.613 

.... 

200.0 

j  B.  M.  on  root  of  L.  O.  tree  60'  to 
1  right  of  line. 

0 

.... 

2.3 

203.3 

1 

0.8 

204.8 

2 

5.7 

199.9 

3 

7.8 

197.8 

4 

9.9 

195.7 

© 
5 

1.120 

196.310 

10.423 
6.3 

195.190 
190.0 

1  On  peg  at  4  -f  30'  -  20'  to  left  of  line, 
J  by  small  P.  O.  tree. 

6 

4.5 

191.8 

Here  the  elevation  of  the  datum  was  taken  200.  00  feet  below 
the  first  bench-mark.     The  instrument  was  set  up  near  Station  2, 


PRELIMINARY    SURVEYS.  29 

and  a  reading  of  5.613  taken  on  the  bencb;  this  was  written  in 
the  B.  S.  column,  and  when  added  to  the  elevation  of  the  bench 
gives  the  height  of  instrument,  205.613.  A  reading  of  2.3  was 
taken  on  Sta.  0,  recorded  in  the  F.S.  column,  and  when  sub- 
tracted from  the  H.I.  yields  an  elevation  of  203.3.  The  eleva- 
tions of  other  points  were  determined  in  the  same  way.  A  little 
beyond  Station  4  the  rod  man  drove  a  peg  and  held  the  rod  on  it, 
yielding  a  reading  of  10.423  and  an  elevation  of  195.190.  The 
instrument  was  then  moved  to  a  point  near  Station  7  and  a  read- 
ing of  1.120  taken  on  the  peg;  this  added  to  195.190  made  the 
new  II.  I.  196.310,  and  the  process  continued  with  this  H.  I. 

In  most  cases  it  will  be  sufficient  to  read  benches  and  turning- 
points  to  hundredths  and  intermediate  points  to  tenths. 

It  will  be  seen  from  the  notes  that  any  error  in  a  turning-point 
causes  the  same  error  in  all  succeeding  points.  To  guard  against 
this  the  roclman  is  required  to  keep  a  "  peg-book,"  in  which  the 
heights  of  instrument  and  elevations  of  turning-points  are  re- 
corded, and  which  must  check  with  the  leveler's  record. 

57.  Wind  and  sunshine  affect  the  accuracy  of  the  work  with 
the  level,  as  is  also  the  case  with  the  transit.  For  very  great 
accuracy  a  calm,  cloudy  day  is  the  best,  but  the  railroad  engineer 
cannot  always  choose  the  best  times  for  his  work,  and  must  take 
such  precautions  as  may  be  possible  while  he  exercises  the  great- 
est care  to  prevent  and  detect  errors.  The  adjustments  should 
be  tested  at  least  once  a  week,  even  when  the  greatest  care  has 
been  taken,  for  unequal  expansion  and  other  causes  may  con- 
spire to  cause  them  to  change. 

By  making  foresights  and  backsights  to  turning-points  about 
equal  the  error  due  to  curvature  will  be  eliminated;  the  readings 
of  rodman  at  these  points  should  also  be  checked.  The  roclman 
should  hold  his  rod  vertical,  which  is  sometimes  accomplished 
by  means  of  a  level  attached  to  rod;  or  the  leveler  can  tell  by  his 
vertical  wire  when  the  rod  is  in  the  same  vertical  plane  with  the 
instrument,  and  by  causing  the  rodman  to  wave  his  rod  back  and 
forth  slowly,  after  clamping  the  target,  he  can  tell  if  the  hori- 
zontal wire  just  bisects  the  target  at  its  highest  position. 

58.  The  Rod  should  be  graduated  to  feet  and  tenths,  reading 
by  target  at  turning-points  and  benches;  intermediate  readings 
are  made  by  the  leveler  at  his  instrument.  Strength  and  dura- 
bility are  essential  qualities.  The  Philadelphia  rod  seems  to 


30        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

answer  the  purpose  as  well  as  any  other  now  manufactured;  the 
Troy  rod  may  be  used  in  the  same  manner  as  the  Philadelphia 
rod,  but  is  lighter  and  less  able  to  stand  rough  usage. 

ARTICLE  5.  THE  TOPOGRAPHIC  PARTY. 

59.  The  Topographic  Party  follows  the  level  and  secures  all 
the  data  necessary  for  making  an  accurate  contour-map  of  a  strip 
of  country  extending  as  far  each  side  of  the  preliminary  as  may 
be   needed   for   the   intelligent   projection   of  the   location-line. 
This  distance  may  vary  from  50  to  300  or  400  feet,  its  width  de- 
pending on  the  difficulties  to  be  encountered  and  the  degree  of 
precision  with  which  the  preliminary  approximates  to  the  final 
location-line.      The  lateral  slope  of  surface  is  obtained  at  the 
stations  of  preliminary  by  means  of  the  hand-level  and  tape,  by 
the  slope-level  or   clinometer,  by  cross  section   rods,  or   by  the 
transit  and  stadia.     Strictly  speaking  the  topography  includes  all 
the  surface  features,  but  for  railroad  work  the  surface  elevations, 
streams,  and  nature  of  surface  are  the  most  important;  it  may 
be  necessary  to  note  the  positions  of  roads,  buildings,  etc.,  and 
should  always  be  done  when  practicable  without  undue  loss  of 
time.     A  pocket-compass  will  be  of  use  in  observing  the  bear- 
ings of  lines. 

60.  There  are  two  methods  of  recording  the  data  obtained; 
one  by  means  of  notes  and  sketches  in  a  book,    the  other  by 
drawing  the  contours  directly  on  the  field-sheet  as  the  data  are 
obtained.   Station  elevations  can  be  taken  direct  from  the  leveler'g 
notes,  and  constitute  the  base  on  which  the  contour  elevations 
rest. 

Suppose  the  hand-level  to  be  used  and  the  notes  kept  in  a  book, 
to  be  afterwards  transferred  to  the  map.  Starting  with  the 
known  center  elevation,  the  topographer  notes  the  height  of  his 
eye  above  the  ground  and  calculates  the  height  of  center  above 
or  below  the  next  contour;  from  this  the  reading  of  the  rod  when 
held  on  this  contour  is  found,  being  the  height  of  station  above 
contour  plus  the  height  of  eye.  He  directs  the  slopeman  in  or 
out  on  a  line  at  right  angles  to  preliminary  until  this  reading  is 
given  by  the  hand-level;  the  distance  out  is  then  measured  and 
recorded,  just  as  in  setting  slope-stakes,  and  the  slopeman  di- 
rected into  position  on  the  next  contour,  in  the  same  manner. 

Thus  if  5-foot  contour-intervals  are  employed,  and  the  station 


PRELIMINARY 


UNIVERSITY 

RVEY33= 


31 


elevation  is  321.  6  feet  und  the  height  of  eye  5.3  feet,  we  shall  have 
for  the  reading  at  the  320-foot  contour  5.3  +  (321.6  —  320)=  6.9. 
Motion  the  slopeman  down  the  slope  until  his  rod  reads  6.9  and 
measure  the  distance  out,  suppose  21  feet.  The  315-foot  contour 
will  be  5  feet  lower,  giving  a  reading  of  11.9,  which  may  be 
found  in  like  manner  at,  say,  80  feet  out.  As  the  rod  reads  only 
to  about  12  feet  the  topographer  must  move  out  to  this  last  point, 
and  with  the  reading  5.3  -f-  5=  10.3  find  the  310-foot  contour  in 
the  same  way.  On  the  up-hill  side  the  325-foot  contour  will  be 
found  with  a  reading  of  5.3  -  (325  -  321.6)  =  1.9  feet,  and  other 
contours  in  like  manner. 

The  notes  may  be  written  thus 


Sta. 

Left. 

Center  Elev. 

Right. 

824 

305   310   315   320 
T93'  125'   80'   2l 

321.6 

325  330   335   340 
27  '  "56"'  "80"'  112 

The  number  above  the  line  is  the  contour  elevation,  the  num- 
ber below  its  distance  out  from  center. 

If  preferred  the  elevation  can  be  taken  at  regular  distances  out 
and  recorded  as  above;  the  position  of  the  contour  will  then  be 
fouud  by  interpolation  when  mapping  the  work. 

61.  If  the  topography  is  to  be  plotted  in  as  the  work  progresses 
the  topographer  must  have  a  light  drawing-board  with  a  pocket 
and  flap  on  back  for  holding  the  sheets  on  which  the  transit-line 
has  been  plotted  the  night  before  ;  the  station  elevations  are 
marked  on  the  line  and  the  contour  positions  spotted  in  as  ob- 
tained by  slopemen,  after  which  the  contours  are  sketched  in. 
Points  where  contours  cross  transit-line  are  found  in  the  same 
manner  as  side  points.  The  size  of  the  sheets  will  depend  on  the 
taste  of  topographer  and  size  of  drawing-board;  17x24  to  19x28 
inches  are  good  sizes. 

The  topographer  will  soon  learn  to  guess  at  the  position  his 
contours  will  occupy  at  the  next  station  ahead,  and  will  sketch 
them  in  lightly,  to  be  erased  and  corrected  when  necessary.  It  is 
often  sufficient  to  take  lateral  readings  at  every  second  or  third 
station. 


62,  If  the  Slope  level  is  used,  the  inclination  of  the  surface  is 
obtained;  then   by  the  use  of  a  scale  constructed  to  show  the 


32        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS.  * 

horizontal  distance  apart  of  contours,  for  the  given  contour-in- 
terval, for  slopes  varying  from  1°  to  20°,  the  position  of  contours 
can  at  once  be  spotted  on  the  map.  Wellington  recommends  the 
use  of  the  altazimuth  as  permitting  the  employment  of  either 
method  at  will — the  altazimuth  being  merely  a  hand-level  with 
a  clinometer  attached. 

63.  Cross-section  Rods  are  measuring- rods  10  or  12  feet  long 
'carrying  a   level-bubble.     By   placing  one  end  at   the   center, 

bringing  the  rod  horizontal,  and  noting  the  height  of  the  end  of 
rod  on  the  down-hill  side,  the  slope  may  readily  be  obtained  and 
the  contours  worked  in  as  before.  For  very  rough,  broken 
ground  this  method  may  be  preferable  to  either  of  the  others. 

64.  If  the  Transit  and  Stadia  are  employed,  very  elaborate 
topography  may  be  taken  with  very  little  field-work,  but  the  ob- 
servations require  considerable  reduction.     With  a  suitable  topo 
graphic  protractor  and  the  slide-rule  mentioned  in  33,  the  large 
number  of  points  that  may  be  obtained  from  each  setting  of  the 
transit  may  be  readily  plotted  and  their  elevations  marked  on  the 
plot,  after  which  the  contour-lines  can  be  worked  in,  and  other 
features  mapped.     For  small  vertical  angles  no  horizontal  reduc- 
tion is  needed. 

While  not  generally  favored  by  railroad  engineers  in  the  past, 
this  method  is  probably  the  most  rapid  and  economical  of  any  so 
far  employed  in  topographic  work. 

ARTICLE  6.  PRELIMINARY  ESTIMATES. 

65.  After  completing  the  field-work  of  the  preliminary  survey 
the  party  is  usually  disbanded,  only  the  transitman,  leveler,  and 
topographer  being  retained  to  assist  the  chief  of  party  to  complete 
the  map,  profile,  and  estimate  of  cost. 

66.  The  Map  may  be  drawn  to  any  suitable  scale,  but  less  than 
400  feet  to  the  inch  is  not  to  be  recommended  where  it  must  be 
used  in  projecting  location.     The  transit-line  is  laid  down  first 
and  the  topography  worked  in  afterwards  from  the  field-map  or 
topographer's  notes.     If  it  is  wanted  on  a  continuous  sheet,  the 
transit-line  must  first  be  drawn  on  a  succession  of  small  sheets, 
which  are  added  as  the  plotting  progresses,  a  new  sheet  being 
slipped  under  the  edge  of  the  preceding  and  tacked  down  when 


PRELIMINARY    SURVEYS.  33 

required.  The  overlapping  edge  is  marked  by  a  number  of  short 
Hues  extending  over  onto  the  sheet  beneath,  to  enable  one  to  re- 
place in  the  proper  position.  When  the  line  has  been  plotted  the 
sheets  are  pasted  together  and  the  whole  shifted  so  as  to  bring  the 
transit-line  over  the  continuous  sheet.  Angular  points  are  then 
pricked  through  and  the  line  drawn  on  the  continuous  sheet. 
Ordinarily  it  will  answer  to  have  the  map  drawn  on  a  succession 
of  small  sheets,  to  be  joined  together  as  required. 

The  plotting  had  best  be  done  by  bearings,  though  it  may  be 
done  from  the  deflection  angles,  provided  care  is  used  to  check 
frequently  by  bearings.  Otherwise  an  error  in  one  angle  wilL 
throw  all  the  remaining  portion  of  the  line  out  of  position. 

If  more  than  one  preliminary  was  run,  they  should  all  be  shown 
on  the  same  sheet  whenever  possible. 

67.  The  Profile  will  be  drawn  by  the  leveler  on  profile-paper, 
and  shows  a  developed  vertical  projection  of  the  line.     The  scale 
will  depend  on  the  paper  used.     There  are  three  scales  in  general 
use,    styled   respectively  Plates  "A,"  "B,"  and  "  C."    There  is 
also  a  metric  profile-paper.     Plate  "  A  "  has  the  vertical  exagger- 
ated 20  to  1  as  compared  with  the  horizontal  and  is  the  best  to 
use  where  much  rockwork  is  expected.     The  vertical  exaggera- 
tion of  Plate  "  B"  is  less  than  of  Plate  "A  ";  this  plate  is  most 
used  for  ordinary  earthwork. 

A  strip  of  color  laid  on  below  the  surface-line,  and  fading  out 
at  the  lower  edge,  adds  greatly  to  the  appearance  of  the  profile. 
The  tentative  grade-line  and  points  of  change  should,  be  drawn  in 
red. 

68.  Preliminary  Estimates  of  quantities  are  made  by  assuming 
a  grade-line  and  drawing  it  on  the  profile;  then  the  cuts  and  fills 
are  taken  from  the  profile,  and  the  corresponding  quantities  ob- 
tained from  Table  XIX  for  the  base  the  road  is  intended  to  have 
when  completed.     The   nature  of  the  work,  whether  ordinary 
earth  or  rock,  can,  of  course,  be  only  roughly  estimated. 

Bridging  is  estimated  from  the  profile  where  piling  or  framed 
bents  may  be  used,  but  where  piers  and  long  spans  are  needed 
special  surveys  with  soundings  are  required,  Culverts,  drains, 
cattle  guards,  cross-ties,  and  rails  for  main  line  and  sidings, 
switch  stands,  buildings,  right  of  way,  clearing,  and  other  factors 
entering  into  the  question  of  cost  must  all  be  considered  aud 
allowed  for  in  making  up  the  estimate. 


34        A   FIELD-MANUAL   FOE   RAILROAD    ENGINEERS. 

Engineering  expenses  and  unforeseen  outlays  that  are  sure  to 
arise  should  have  a  liberal  allowance. 

69.  The  Report  of  the  chief  of  party  should  set  forth  the  ad- 
vantages and  probable  cost  of  each  of  the  several  lines  run 
when  there  is  more  than  one.  On  this  report  frequently  depends 
whether  or  not  the  line  is  to  be  located,  and  it  should  be  clear 
and  exhaustive,  though  plainly  and  concisely  worded.  The  map 
and  profile  form  an  integral  part  of  the  report  and  show  from 
what  data  the  estimates  were  derived. 


CHAPTER  III. 

LOCATION. 

AUTICLE  7.     PROJECTING  LOCATION. 

70.  After  the  preliminary  lias  been  mapped  and  the  topography 
worked  iu,  the  engineer  proceeds  to  make  a  paper  location  for  his 
guidance  in  the  field.     The  solution  of  the  varied  and  complex 
problems  that  confront  him   are  more  or  less  interdependent. 
The  guiding  principle,  applicable  to  all  departments  of  engineer- 
ing, that  the,  best  structure  is  that  which  for  the  least  cost  best  an- 
swers the  purpose  for  which  it  was  intended,  should  control,  even 
though  the  resulting  structure  be  inferior,  in  point  of  scientific 
design,  to  some  other.     The  best  road  as  regards  construction  and 
grades  may  be  a  failure  because  of  excessive  first  cost,  while 
the  cheapest  construction  will  entail  such  heavy  operating  ex- 
penses that  it  may  b'e  equally  unprofitable.     The  alignment  must 
be  as  free  from  curves  as  possible,  while  heavy  grades  are  at  the 
same  time  excluded;  these  two  requirements  conflict  and  must 
be  as  well  adjusted  as  possible.     The  amount  of  earthwork,  of 
bridging  and  other  structures  must  be  kept  down  to  the  lowest 
limits. 

71.  Starting  at  the  summit  of  the  most  difficult  portion  of  the 
route,  assume  a  starting-point  and  elevation;  with  the  dividers  set 
at  such  a  distance  to  the  scale  of  the  map  as  will  give  a  fall  of  one 
contour-space — or  half  space— for  the  assumed  grade,  step  down 
the  slope  in  such  a  way  that  the  dividers  fall  each  time  on  the 
next  lower  contour,  or  half-space,  according  to  the  fall  assumed  in 
setting  dividers.     If  curve  compensation  is  allowed,  the  dividers 
must  be  reset  for  each  curve,  for  the  same  fall,  since  the  grade 
will  be  slackened  on  curves.     The  points  at  which  the  dividers 
fall  are  lightly  spotted  on  the  map  and  connected  by  a  grade 
contour,  which   represents  the  surface-line  having  the  required 
gradient.     This  line  will  be  too  broken  to  be  used  as  a  location- 

35 


36        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

line,  so  we  have  then  to  draw  on  the  map  a  succession  of  curves 
aud  tangents  that  will  approximate  sufficiently  close  to  it,  at  the 
same  time  that  a  proper  balance  is  maintained  between  earthwork 
and  curvature. 

Having  lightly  plotted  the  proposed  line,  the  elevations  are 
transferred  to  profile-paper,  thus  giving  a  profile  of  the  line. 
With  a  fine  thread  stretched  along  the  profile,  to  represent  the 
grade-line,  adjust  the  cuts  and  fills  to  suit  the  nature  of  the  work. 
In  general,  fills  are  cheaper  than  cuts  both  in  construction  and 
maintenance;  and  especially  is  this  true  where  a  shallow  surface 
layer  of  earth  is  underlaid  by  rock.  It  may  happen  that  the 
material  from  excavation  must  be  used  in  embankment,  when 
the  cuts  and  fills  must  be  made  to  balance  by  shifting  the  grade- 
line  until  this  appears  to  be  the  case  on  the  profile. 

At  the  stream  crossings  the  grade-line  must  be  kept  safely 
above  high-water  mark,  so  that  sufficient  waterway  is  provided, 
and  allowance  made  therefor. 

After  locating  the  most  difficult  portions  pass  on  to  the  easier 
work,  returning  later  on  to  study  the  effect  this  will  have  on  the 
part  first  located.  It  may  be  necessary  to  go  over  the  projection 
several  times  before  you  can  be  reasonably  sure  that  the  best  loca- 
tion has  been  projected;  even  then  the  study  of  the  liue  in  the 
field  will  cause  many  of  the  details  to  be  altered,  sometimes 
materially. 

Long  grades  are  to  be  preferred  to  short  ones,  but  questions  of 
economy  may  necessitate  the  latter  in  order  to  lighten  work;  care 
must  be  taken  that  the  grades  are  not  so  badly  "  chopped"  that 
they  interfere  with  the  easy  riding  of  the  train. 

In  projecting  the  line  it  will  generally  be  best  to  strike  the 
curves  first  and  draw  the  tangents  afterwards,  though  it  some- 
times happens  that  long  tangents  will  control  the  curves;  when 
this  is  the  case  the  tangents  are  drawn  to  intersection  and  the 
curves  afterwards  put  in. 

When  transition-curves  are  employed,  a  slight  offset  should  be 
made  at  the  beginning  and  end  of  curves  to  allow  for  their  inser- 
tion in  the  field.  These  offsets  will  be  so  small  that  it  is  useless 
to  attempt  to  show  them  to  scale. 

72.  A  Curve-protractor  will  be  of  material  assistance  in  find- 
ing the  degree  of  curve  required  to  unite  two  tangents  that  have 
been  laid  down  on  the  map.  It  consists  of  a  transparent,  semi- 
circular protractor  having  a  series  of  curves  from  30'  up  to  8° 


LOCATION.  37 

plainly  cut  upon  it.  The  curves  are  on  both  sides,  those  on  the 
reverse  side  having  their  concavities  turned  in  an  opposite  sense 
from  those  on  the  face.  The  scale  is  usually  400  feet  to  the  inch, 
and  in  any  case  the  map  and  protractor  must  be  drawn  to  the 
same  scale.  Sometimes  a  set  of  cardboard  or  hard-rubber  curves 
are  used,  but  they  are  inferior  to  the  curve-protractor.  To  use 
it,  simply  prolong  tangents  to  intersection  and  then  place  the 
protractor  so  that  the  curve  admitting  of  the  best  grade  is  tan- 
gent to  the  two  straight  lines.  Mark  the  points  of  taugency, 
which  will  be  the  beginning  and  end  of  curve.  When  the  curve 
is  required  to  pass  through  a  given  point  the  proper  curve  may 
be  immediately  found  by  trial,  whereas  the  calculations  would 
require  some  little  time. 

Reversed  curves  should  never  be  allowed  on  main  lines.  Suffi- 
cient tangent  should  be  interposed  to  allow  space  for  easing  off 
the  superelevation  of  outside  rails,  or  for  the  insertion  of  tran- 
sition-curves when  these  are  to  be  employed. 

73.  The  Field  Corps  is  substantially  that  required  on  the  pre- 
liminary survey,  and  the  methods  of  work  pretty  much  the  same, 
except  that  curves  must  now  be  run  in,  and  this  necessitates  more 
clearing.  If  first  and  second  location-lines  are  to  be  run  (and  it  is 
real  economy  to  run  both),  it  will  not  be  necessary  to  have  the 
stationing  continuous  on  the  first,  so  the  pluses  arising  from 
"  backing  up"  need  only  be  noted  and  eliminated  when  the  final 
location-line  is  run.  If  transition-curves  are  to  be  inserted,  they 
need  not  be  run  the  first  time,  the  proper  offset  being  made  at 
the  P.  T.  or  P.  C.  of  the  circular  curves,  which  latter  are  to  be  run. 

On  the  final  location-line  the  stationing  must  be  continuous, 
beginning  with  zero.  The  stakes  are  marked  as  on  the  pre- 
liminary survey,  and  all  hubs  that  are  likely  to  be  used  again 
must  be  referenced  in,  the  reference-hubs  being  set  well  out  of 
the  way  of  disturbance  by  the  plow  or  scraper. 

The  leveler  should  make  bench-marks  every  1000  or  2000  feet, 
to  be  used  in  running  check-levels  and  in  giving  grades  later  on. 

From  the  paper  location  the  notes  should  be  made  up  in  the 
office,  to  serve  as  a  guide  in  the  field;  however,  no  attempt  should 
be  made  to  adhere  rigidly  to  them,  since  slight  errors  in  the 
mapping  will  affect  the  projected  line,  while  in  the  field  the  line 
may  be  shifted  here  and  there  so  as  to  fit  the  ground  more  snugly 
and  accord  more  closely  with  what  the  nature  of  the  earthwork 
demands. 


B8        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

The  highest  skill  of  the  engineer  is  required  to  secure  the  best 
location-line,  and  he  should  have  all  the  time  he  needs.  Undue 
baste  on  location— as  on  reconuoissance  and  preliminary— is 
almost  sure  to  result  in  increased  cost  of  construction. 


ARTICLE  8.    SIMPLE  CURVES. 

A.  Definitions  and  Formulas. 

74.  The  Circular  Curves  that  are  usually  employed  to  unite 
straight  reaches  of  the  railroad  may  be  simple,  compound,  or  re- 
versed. The  use  of  reversed  curves  should,  however,  be  limited 
to  turnouts  and  cross-overs. 

a.  A  Simple  Curve  is  the  arc  of  a  circle. 

b.  A  Compound  Curve  consists  of  two  simple  curves,  of  differ- 
ent radii,  both  on  the  same  side  of  a  common  tangent. 

c.  A  Reversed  Curve  is  made  up  of  two  curves  of  contrary 
flexure  having  the  same  or  different  radii,  and  a  common  tangent. 

d.  The  Point  of  Curve  (P.O.)  is  the  end  of  tangent  and  begin- 
ning of  curve,  as  at  A,  Fig.  12. 


FIG.  12. 

e.  The  Point  of  Tangent  (P.T.)  is  the  end  of  curve  and  be- 
ginning of  tangent,  as  at  B  of  Fig.  12. 

/.  The  Point  of  Intersection  (P.I.)  is  the  point  where  the 
tangent  at  the  P. C.  and  P.T.  intersect  when  produced.  (D  of 
Fig.  12.) 

g.  The  Intersection  Angle  (T\  is  the  angle  at  the  P.I.  be- 
tween the  tangents  meeting  there,  and  equals  the  angle  at  the 
center. 

h.  The  Tangent  Distance  (T)  is  the  length  of  the  produced 
tangent  measured  from  the  P.  C.  or  P.  T.  to  the  P.I.  The  term 


LOCATION. 


39 


tangent  is  applied  to  any  straight  portion  of  the  line,  but  the  letter 
T  will  be  used  to  designate  the  produced  portion  only. 

*".  The  Mid-ordinate  (M)  is  the  portion  of  the  radius  inter- 
cepted between  the  arc  and  chord  when  it  cuts  the  chord  at  its 
middle  point. 

j.  The  External  (E)  is  the  part  of  the  radius  produced  to  the 
P.L,  intercepted  between  curve  and  the  P.I. 

k.  The  Long  Chord  (L.C.)  is  the  chord  joining  the  P.  C.  and 
P.  T.  Frequently  the  term  is  applied  to  any  chord  longer  than 
the  unit  chord. 

I.  The  Radius  will  be  denoted  by  R. 

in.  The  Point  of  Compound  Curve  (P.  G.  C. )  is  the  point  of 
common  tangency  of  the  two  branches  of  a  compound  curve. 
(See  Fig.  13.) 


n.  The  Point  of  Reversed  Curve  (P.R.C.)  is  the  point  of 
common  tangeucy  of  the  two  branches  of  a  reversed  curve. 

o.  The  Degree  of  Curve  (JO)  is  the  angle  at  the  center  sub- 
tended by  the  unit  chord.  In  the  United  States  this  chord  is  100 
feet,  in  England  66  feet,  and  where  the  metric  system  is  em- 
ployed it  is  taken  at  20  meters.  Any  convenient  chord  length 
may  be  taken,  but  for  uniformity  American  engineers  have 
adopted  the  chord  of  100  feet,  and  unless  otherwise  stated  it  is 
always  so  understood  when  we  speak  of  the  degree  of  curve. 

Half  the  degree  of  curve  is  called  the  deflection-angle,  since 
it  is  the  angle  to  be  deflected  from  the  tangent  to  the  chord. 

If  there  were  any  practical  method  of  measuring  around  the 
curve  instead  of  along  the  chord,  an  accurate  and  convenient 
ratio  for  expressing  the  radius  in  terms  of  the  degree  would  be 
hud.  Thus  if  D  is  the  angle  at  the  center  subtended  by  the  arc 
of  unit  length,  we  have,  where  a  is  this  unit  arc, 


40        A   FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


Hence 


. 


a     360 
'        ' 


When  a  equals  100  ft.  this  becomes 
100    360 


(11) 


(HO 


R  varies  inversely  as  D,  so  that  knowing  the  radius  for  a  1° 
curve,  we  should  have  only  to  divide  this  by  D  to  get  the  radius 
for  a  D°  curve. 

Since  the  chord  is  employed  instead  of  the  arc,  we  determine 
H  by  means  of  the  following  problem 

75.  Given  the  Chord  C,  and  Degree  of  Curve  D,  to  Find  the 
Radius  R. 

In  Fig.  14,  AB  is  the  chord  C,  OE  a  perpendicular  from  the 
center  upon  AB 


From  the  right  triangle  AEO  we  have 
R  sin  \V  =    C. 


Whence 


When  (7  is  100  ft., 


50 


(12) 


=  50  cosec  \D (120 


LOCATION.  41 

Comparing  results  given  by  formula  (12')  with  those  given  by 
(IT),  we  have  for  a  few  curves: 

Degree  of  Curve.  R  by  (12').  -K  by  (IT).  Difference. 

1  ............  5729.65  5729.58  0.07 

2  ............  2864.93  2864.79  0.14 

3  ............  1910.08  1909.86  0.22 

5  ............  1146.28  1145.92  0.36 

7  ............  819.02  818.51  0.51 

10  ............       573.69  572.96  0.73 

14  ............      410.28  409.26  1.02 

20  ............      287.94  286.48  1.46 

The  difference  is  seen  to  be  about  one  half  a  foot  for  a  7° 
curve,  one  foot  for  a  14°  curve,  and  one  and  one-half  feet  for  a 
20°  curve. 

Up  to  a  7°  curve  the  difference  is  inconsiderable,  and  we  may 
stake  out  curves  with  100-  foot  chords.  From  7  to  14  degrees  50- 
foot  chords  may  be  used.  Therefore 


"        =    cosec 


For  curves  from  14°  to  28°  we  should  use  25-foot  chords, 
for  which 


=  12.5  CoSec  ID.  (12'  b) 


Above  28°  shorter  chords—  say  10  feet—  should  be  used,  if  the 
curve  cannot  be  struck  from  the  center.     In  this  case 


Table  I  of  radii  was  computed  by  formulas  (12'),  (12'a),  and 
(12'ft). 

In  practice  it  is  customary  to  take  the  radius  of  a  1°  curve  as 
5730  feet  and  to  assume  the  radii  to  vary  inversely  as  the  degree  ; 
thus  for  a  4°  curve  the  radius  would  be  R=  *J4£  =  1432.5  feet, 
while  by  Table  I  it  is  1432.69  feet—  a  difference  of  only  .19  foot  ; 
for  a  12°  curve  JR  =  ^jp  =  477.5  feet,  while  by  Table  I  it  is 
477.68  feet.  The  effect  of  taking  5730  instead  of  5729.65  for  the 
radius  of  a  1°  curve  is  to  reduce  the  error  resulting  from  the 
assumption  that  11  equals  5730  divided  by  the  degree  of  curve. 


42        A    FIELD-MANUAL    FOR    RAILROAD    ENGINE K US. 

76.  The  Length  of  Curve  (L)  is  found  by  dividing  the  angle 
at  the  center  (which  equals  the  intersection)  angle)  by  the  degree 
of  curve,  the  result  being  in  chains  and  decimals  of  a  chain.    The 
number  of  P.  G.  -j-  L  will  give  the  station  number  of  P.  T. 

EXAMPLE.— The  P.  C.  of  a  4°  curve  having  /  =  26°  30'  is  at  sta. 
104  + 12.5.  Find  L  and  the  number  of  the  P. T.  Here 

no  K 

L=  -^  =  6. 625  chains. 

104.125  -f-  6.625  =  110.75  ;  hence  the  number  of  P.  T.  is 
110  +  75. 

77.  Use  of  the  Table  of  Functions  of  a  One-degree  Curve. — 

In  the  location  of  railway  curves  geometrical  accuracy  will 
frequently  be  of  less  importance  than  rapidity  of  field-work,  so 
long  as  errors  are  kept  within  certain  limits. 

On  tangents  slight  errors  of  alignment  may  readily  be  detected 
by  the  unaided  eye,  but  on  curves  these  are  not  so  apparent. 
Moreover  it  is  not  likely  that  the  trackmen  will  keep  them  up  in 
the  exact  position  of  their  location. 

To  simplify  and  shorten  the  field  computations  engineers  make 
use  of  a  table  of  functions  of  a  1°  curve,  and  assume  these  func- 
tions for  other  curves  to  vary  inversely  as  their  degree,  or  directly 
as  their  radii.  Table  IX  gives  values  of  the  tangent  distances, 
long  chords,  mid-ordinates,  and  externals  for  a  1°  curve,  the 
radius  of  which  is  taken  as  5730  feet.  To  find  these  functions 
for  other  curves,  divide  the  tabular  values  by  the  degree  of  curve. 
The  error  resulting  from  this  assumption  will,  in  any  practical 
case,  amount  to  no  more  than  a  few  tenths  or  huudrcdths  of  a 
foot. 

Table  IX  may  also  be  used  as  a  metric  curve  table,  the  tabular 
values  being  taken  as  meters  instead  of  feet,  If  the  unit  metric 
chord  is  20  meters  long,  this  may  be  taken  as  one  fifth  of  the 
tabular  unit  chord;  so  to  use  the  table  multiply  the  metric  degree 
by  5  and  enter  the  table  with  the  result  as  a  value  of  D. 

For  instance,  a  2°  metric  curve  having  7  =  40°  would  have  a 

Q/i  £\  f* 

mid-ordinate  equal  to  „-— ~  =  34.56  meters. 
£  X  5 

For  the  approximate  radius  of  a  metric  curve  divide  5780  by  5 

57SO 
times  the  degree.     Thus  a  4°  metric  curve  would  have  R=  •£• 


LOCATION.  43 

—  286.5  meters.     For  the  exact,  radius  m.ike  use  of  formula  (12). 
Thus  for  a  4°  curve  having  20-meter  chords  R  =  — — 53  =  286.54 

meters,  a  difference  of  only  .04  meters. 

If  a  metric  curve  is  to  be  retraced  with  a  100-ft.  chain,  we 
convert  the  metric  degree  to  the  degree  referred  to  100-ft.  chords 
by  the  relation  that  a  100-ft.  chain  =  1.524  chains  of  20  meters 
each;  a  20-meter  chain  =  65.618  ft.;  one  foot  =  0.3048  meters; 
one  meter  =  3.2809  ft. 

It  will  sometimes  be  a  sufficiently  close  approximation  to  take 
the  20  meter  chain  as  two  thirds  of  a  100-ft.  chain;  this  will  make 
the  metric  curve  nearly  two  thirds  of  the  degree  the  same  curve 
would  have  when  laid  out  with  a  100-ft.  chain,  and  the  curve  with 
100-ft.  chords  nearly  three  halves  of  the  degree  as  laid  out  with 
the  20-meter  chain.  Thus  a  4°  metric  curve  would  be  equivalent 
to  a  6°  curve  laid  out  with  a  100-ft.  chain. 

In  the  problems  that  follow  two  methods  of  solution  will  be 
given  when  practicable — the  first  being  rigid,  while  the  second 
is  based  on  the  use  of  Table  IX.  To  shorten  the  formulas  the 
subscript  1  will  be  written  after  the  letters  T,  L.  0.,  M,  and  E 
when  these  are  the  functions  of  a  1°  curve.  Thus  Ti  ^  28°  means 
the  tangent  distance  for  a  1°  curve  when  7=28°,  L.C.i  ^  16° 
the  long  chord  for  a  1°  curve  when  /=  16°,  etc. 

78.  Tables  of  Natural  and  Logarithmic  Circular  Functions.  — 
Many  engineers  prefer  to  work  altogether  by  tables  of  natural 
sines,  cosines,  etc.,  and  time  may  often  be  saved  by  their  use. 
Nevertheless  logarithmic  tables  are  of  frequent  advantage,  even  in 
the  field,  and  the  more  important  ones,  such  as  the  logarithmic 
sines,  cosines,  tangents,  and  cotangents,  together  with  the  loga- 
rithms of  numbers,  are  given  in  the  back  of  the  book  along  with 
the  tables  of  natural  functions. 

79.  Given  7?  and  C  to  Find  Z>. 

From  equation  (12), 

sin  \D  =  i? (13) 

80.  Given  7  and  R  (or  D)  to  Find  T. 

If  I)  is  given,  find  R  by  (12');  then  in  Fig.  15  from  triangle 
OAB  we  get 

T  =  K  tan  11. (14) 


44        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

BY  TABLE  IX.     Find  the  tabular  value  of  T  for  the  given 
angle  /;  then 

(Ha) 


EXAMPLE.—  7=  35°  40',  D  =  4°;  required  T. 
By  (14),     T—  1432.69  tau  17°  50'  -  460.91  feet. 


By  (14a),  T=     ~       =  460.85  feet,  a  result  differing  from  the 
value  found  by  the  rigid  method  by  only  0.06  foot. 

81.  Given  /  and  Tto  Find  ft  or  D 

From  (14), 


tan 


=  TcotU.     .....    (15) 


Then  by  Table  I  the  degree  may  be  found. 
BY  TABLE  IX. 


(15a) 


82.   Given  J  and  D  to  Find  the  Long  Chord  L.  C. 
First  find  R  by  (12)  or  (12'),  or  by  Table  I  ;  then  from  the 
triangle  OA  F  of  Fig.  15, 


(16) 


LOCATION. 


45 


BY  TABLE  IX. — Find  the  tabular  L.  (7.  for  the  given  angle  /; 
then 


L.C.= 


D 


83.  Given  the  Radius  R  and  any  Chord  C  to  Find  the 
Ordinate  to  the  Curve  at  any  Point. 

FIRST  METHOD.— In  Fig.  16  let  HE  be  the  chord  C\  UK—  a 
and  KE  —  b,  the  segments  into  which  it  is  divided  by  the  ordi- 


nate  y.     Draw  the  radius  through  K;  call  the  portion  between 
chord  and  curve  y'.     By  geometry, 


from  which 


ab 


But  y'  is  small  compared  with  2J?,  and  hence  we  write 


Now  y  does  not  differ  sensibly  from  y'  in  the  cases  met  with  in 
practice,  so  we  write 

ab 

y   rv  ij \"  / 


46        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


If  we  write  R  =         ,  formula  (6)  becomes 

abD 


2  X  5730'    ' 

a  b 

— —  =  m,  T— r  =:  n,  and  substitute  in  (c),  giving 
100  10U 


y  = 

or  very  nearly 


y^lmnD (17) 


y  is  given  in  feet  when  m  and  n  are  in  chains  and  decimals  of 
a  chain. 
At  the  rnid-point  F,  m  —  n,  and  y  =  M . 


(18) 


CAUTION. — Formulas  (17)  and  (18),  while  very  convenient  for 
field  use  in  passing  obstructions,  are  liable  to  error  when  very 
long  chords  or  large  values  of  D  are  used,  since  they  give  results 
that  are  too  small. 

If  we  write  the  arcs  HN,  NE  for  a  and  b,  we  shall  get  results 
that  are  too  large,  yet  about  as  near  the  true  values  as  by  taking 
m  and  n  to  be  the  segments  of  the  chord.  To  illustrate  we  will 
find  a  few  values  of  M  and  compare  with  the  true  values  taken 
from  Table  V. 


Degree 
of 
Curve. 

2       .. 

Length 
of 
Arc. 

2  stations. 

Mid-ord. 
1.75 

Mid-ord. 
by 
M=l(HQ)*D. 

1.75 

Mid-ord. 
by 
Table  V. 

1.75 

2  

6 

15.69 

15.75 

15.09 

5 

2       " 

4.37 

4.38 

4.36 

5  

6       " 

38.51 

39.38 

39.06 

8  
8   

2       " 
4       " 

6.96 
27.29 

7.00 
28.00 

6.97 

27.75 

8 

5       " 

42.02 

43.75 

43.20 

8... 

6       " 

59.43 

63.00 

61.93 

From  this  it  appears  we  may  use  formula  (18)— and  (17)  as 
well — taking  either  the  segments  of  the  arc  or  chord  for  curves 
not  exceeding  4°  with  arcs  up  to  600  ft. ;  for  curves  from  4°  to  6° 


LOCATION.  47 

they  may  be  used  up  to  500-ft.  ares,  \vkile  for  curves  between 
6°  and  8°  not  more  than  400  feet  of  arc  may  be  taken. 

SECOND    METHOD.— First    determine    the    uiid-ordiuate.      In 
triangle  OEF, 


OF= 

then 

$G*.    .....     (19) 


To  find  ordinate  AC  distant  d  from  the  mid-point  of  EH,  draw 
OB  =  d  parallel  to  HE;  draw  AB  at  right  angles  to  HE.     Then 


BA  =  \/IP  -  d*. 
Therefore 


CA  =  y=  VJt2  -  d*  -  VR*  -  iC*.    .     .     .     (20) 

THIRD  METHOD.—  If  the  chord  C  is  short,  we  may  regard  the 
arc  as  an  arc  of  a  parabola,  for  which  it  is  known  that  ordi- 
uates  vary  as  the  product  of  the  segments  into  which  they  divide 
the  chord.  The  mid-ordinate  being  known,  we  have 

y          ab 

Ti  = 


From  formula  (b)  we  have  for  y  —  M,  a  =  b  =  \G, 
M_W_C* 

-~zR-m'   ' 

The  mid-ordinate  for  any  other  chord  C'  is 

M-C" 
M'~SR' 

Hence 

&_&*_ 

M~  C'2' 

.-.  ift.«4f(£-7  .........     (23) 

w  / 

If  C"  =  -|(7,  this  gives 

Ml=\M.    .........     (23') 


48        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

This  last  relation  affords  an  easy  method  of  staking  out  a  curve 
when  the  mid-ordiuate  of  a  given  chord  has  been  determined. 
First  erect  the  ordinate  M  at  the  mid-point  of  the  chord;  then 
join  the  ends  of  chord  with  the  extremity  of  the  ordinate  just 
measured;  the  lengths  of  these  chords  do  not  differ  much  from 
^C;  at  their  mid-points  erect  ordinates  equal  to  \M,  giving  points 
on  the  curve.  Proceed  in  like  manner  for  other  points  until  a 
sufficient  number  have  been  located. 

84.  Given  R  and  /  to  Find  the  External  E. 


In  Fig.  17  E  =  OB  =  OB  -  OG. 
But  OB-R  sec  £/  and  OG  =  R. 


.-.  E  —  .ft(sec  \1  -  1)  = 

BY  TABLE  IX.—  Find  E  for  a  1° 
angle  /;  then 


ex  sec  \L    .    .    .      (24) 
curve  for  an  intersection 


(24a) 


85.  Given  T  and  /  to  Find  E. 

In  Fig.  17  draw  EG  perpendicular  to  AB,  and  produce  AO  to 

B/ 


FIG.  17. 

intersect  EG  at  C.  BCis  parallel  to  AO,  and  the  triangles  AGO 
and  GBC&TG  similar;  hence  BC=BG  =  E.  In  the  right  triangle 
ABC,  angle  BAG—  \BAF  =  $1.  Therefore 

(25) 


E  =  T  tan  \L   .    . 
EXERCISE.— Derive  equation  (25)  from  (24). 


LOCATION". 


49 


86.  Given  M  and  /  to  Find  E. 

From  trigonometry, 

sec  \I  = 
Insert  this  in  (24)  and  we  get 

'  —  cos 


cos 


n 
=  It 


cos  fl 

But  from  Fig.  17,  M  =  R(\  -  cos 

E_     M  • 
~  cos  \I 

87.  Given  E  and  /to  Find  R. 

From  (24), 

E  E 


Substitute  in  (a)  : 


(a) 


11  =  - 


cos 


sec    /  —  1 


ex  sec 


vers 


88.  Given  /  and  E  to  Find  2. 
From  (25), 

r=  ^r 
tan    / 


(26) 


(27) 


(28 


89.  Given  the  Chord  G  and  Degree  of  Curve  D  to  Find 
the  Chord  Deflection  Offset  d. 
In  Fig.  18  extend  EA  to  H,  making  AH  —  EA  =  AB ;  join 


'0 
FIG.  18. 

JET  and  B  and  draw  ^^T  to  the  mid-point  of  HB.     Then 
HK=  EB= 
.-.  d  =  HB  = 


(29) 


50        A   .FIELD-MANUAL    FOR    KAILUOAD    ENGJNEEKS. 

When  C=  100', 

d  =  200  sin  W  ..........      (29') 

1  Q 

If  we  write  sin  \I>  =  ^  from  (12)  in  formula  (29),  there  results 

d  =  ^  .............       (30) 

For  curves  up  to  7°,  G  —  100';  hence 

10000 
<*=  -g--    -    •  _  .........     (30') 

For  curves  from  7°  to  14°,  G  —  50';  therefore 

(30") 
For  #  write       -,  and  (30'),  for  G  =  100,  becomes 


and  for  G  —  50,  (30")  becomes 

ocjnn  T) 

d  =  ^lYD  =  .4363Z>  =  .873.  5-.     .     .     .      (31') 
57oO  £ 

EXAMPLE.—  Fiud  c^  for  a  6°  curve,  G  —  100  feet. 
By  (29'),          d  =  200  X  0.05234  =  10.47  feet. 


By  (30'),          d  =  =  10.47  feet. 

yOO.  4: 

By  (31),  d  =  1.745  X  6  =  10.47  feet. 

90.  Given  the  Chord  G  and  Degree  of  Curve  D  to  Find  the 
Tangential  Deflection  Offset  t. 

In  Fig.  18  make  EF  (tangent  at  E)  equal  to  EA,  and  join  F 
with  A.  Draw  EG  to  the  mid-point  of  FA.  Angle  AEG  = 
GEF  =  \D\  hence,,  from  the  figure, 


(32) 


LOCATION.  51 

When  G  =  100  feet, 

t  =  200  sin  I D (32') 

Since   \D  is  small,  we   may   write,    without   material   error, 
sin  \D  =  i  sin  \D\  then,  writing  sin  \D  —  -^,  as  in  89,  we  get 

t  =  ^ (33) 

Making  G  —  100  ft.  and  writing  R  =  —~  gives 

(33') 


When  C  =  50  feet,  (33)  yields 

t  =  .218Z>  =  .436  X  ~ (33") 

/w 

EXAMPLE.— Find  t  for  a  6°  curve,  C  =  100  ft. 
By  (32')  t  =  200  sin  1°  30'  =  5.24  ft. 

By  (33'),  =  .873  X  6  =  5.24  ft. 

91.  To  Find  the  Subtangential  Deflection  Offset  t'  for  a 
Subchord  C' 

FIRST  METHOD. — By  formula  (13)  find  the  angle  at  the  center 
subtended  by  the  subchord  C';  call  this  angle  I)'.  From  (32), 

t'  =  2C'  sin  \D' (34) 

SECOND  METHOD.— In  Fig.  19,  with  Ens,  center  strike  the  arcs 
FG  and  AH,  taking  EF  =  C'  and  A 

EA  =  C ;  prolong  EG  to  B.    Now 

assuming  that  the  chords  C'  and  C  %          \t 

are  proportional  to   their  central 
angles   we  have 

AB  _t_ 
G'    ~  C '  ' 

From  the  similar   sectors  EFG  FIG.  19. 

and  EAB,  since  EB  =  C, 

C        C' 


52        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 
Multiplying  (a)  and  (b)  together,  term  by  term, 
Gt       C' 


Whence 


(35) 


EXAMPLE.— Find  t'  for  a  7°  curve  when  C  =  60  ft. 


fiO 
.-7- 

i(JO 


Here  IX  =  .-7-  x  7°  (very  nearly)  =  4°  12'. 


By  (34),          t'  =  2  X  60  X  0.01832  =  2.20  ft. 
By  (32)'          t  =  6.11  ft. 

By  (35), 


t'  =  6.11  X  =  2.20  ft. 

uy 


92.  To  Find  the  Tangent  Offset  z. 

In  Fig.  20,  EB  =  zis  the  required  offset.  Let  AE=  n  chains  = 
lOOrc  feet.  AE=FB,  the  half-chord 
having  the  mid-ordhmte  AF  =  .##; 
hence  we  have,  by  formula  (18), 

z  =  ItfD.     .     .     .     (36) 

In  this  formula  we  may  take  n  to 
be  either  the  length  of  AE  or  the  arc 
AB,  in  chains.  If  taken  equal  to  AE 
the  offsets  will  be  slightly  too  small, 
while  if  taken  equal  to  AB  they  will 
be  a  little  too  large.  The  use  of  the 
formula  is  limited  to  small  values  of 
n  and  D,  as  was  pointed  out  in  83. 
(See  CAUTION.) 
Formula  (36)  is  easy  of  application  and  of  frequent  use  in 

locating  curves  by  offsets  from  the  tangents.     For  curves  up  to 

4°   n  may  be  as  great   as  3,  but  for  sharper  curves  it  should 

be  less. 
EXAMPLE.  —  Find  six  offsets  to  a  4°  curve  at  points  50  ft.  apart, 

measured  around  the  curve. 


FIG.  20. 


LOCATION.  53 

By  successive  applications  of  (36)  we  have 

for  n  =  i  z  =  I  X   -1  X  4  =    0.88  feet 

»  =  1,  «  =  |  X    1X4=    3.50     " 

n  =  f,  2  =  I  X    f  X  4  =    7.88     " 

n  =  2,  s  =  |x4x4  =  14.00     " 

n  =  |,  2  =  |  X  -¥-  X  4  =  21.88    " 

n  =  3,  2  =  £X9X4  =  31.50    " 

The  last  value  of  z  is  in  error  by  about  0.2  ft.,  but  for  setting 
stakes  on  construction  this  difference  is  not  material  so  long  as 
the  alignment  beyond  this  point  does  not  depend  on  it.  In 
setting  track-centers  the  completed  road-bed  is  available  and  the 
stakes  may  be  set  with  the  transit,  in  the  usual  way. 

93.  Difference  in  Length  of  a  Circular  Arc  and  its   Long 
Chord. 
FIRST  METHOD. — Let  the  central  angle  be  a  degrees.     By  (13), 

siu  K  =  A 

Changing  degrees  to  circular  measure,  a  (in  n  meas.)  =  — 

-  JL      The  length  of  arc  is  Ra  =  #^-.     Then 
57. o  o7.o 


Arc  -  chord  =  R-—  -c (37) 

o7.o 

SECOND  METHOD. — An  easy  approximation  may  be  found  as 
follows : 

Referring  to  Fig.  17,  AE  =  c,  GF=  M.     Let  A  O  =  b  ^  £-  -f  x. 

a 

From  the  right  triangle  AFO 


TIT2 

From  which  x  —  (a) 

c  -f  x 


54        A    KrEUKMANUAL   FOR   RAILROAD    ENGINEERS. 

Neglecting  the  x  in  denominator  as  small  compared  with  c 
gives 


2  If  2 

Then  will  2b  -  c  -  2x  =  -—  ......     (38) 

c 

From  Huygcns'  approximation  to  the  length  of  a  circular  arc 

(see  Williamson's  Differential   Calculus,  p.  66),  arc  =  —  -f  -  . 

o 

Therefore 

Arc  -  chord  =  ^5-7  -  c  -  |-(2&  -  c).      .     .       (c) 
o 

Inserting  the  value  of  25  —  c  from  (88)  gives 

Arc  —  chord  =  -^—  ..........      (d) 

oC 

When  the  arc  is  not  very  great  we  may  write  c  =  lOOfti  ,  where 
TO  j  is  the  number  of  chains  contained  in  the  arc  AE.  From  (18), 
remembering  that  HI  =  2n, 

M  =  0.218n,9l>. 
Inserting  these  values  of  c  and  M  in  (d), 


Arc  _  cllor<1  =  I  <»i^  =  _Lm,^,  nca,.ly.  .    (39) 

EXAMPLE.  —  Find  the  difference  in  length  of  arc  and  chord  of 
a  4°  curve  when  HI  —  6  stations. 

The  central  angle  is  4x6  =  24n;  then,  from  Table  IV, 
c  =  595.74. 

By  (37), 

24 
Arc  -  chord  =  1432.7  X  ==-    -  595.74  =  4.34  ft. 

O  i  .o 


By  (39), 
An 

REMARK.— Formula  (38)  is  interesting  as  showing  what  a  com- 


.  6X6X6X4X4 

Arc  -  chord  =  -  =  4.32  ft 

ovU 


LOCATION. 


55 


paratively  small  increase  in  length  of  line  is  caused  by  a  consid- 
erable lateral  deflection  in  alignment.  For  instance,  a  lateral 
deflection  of  2000  feet  is  made  at  the  mid-point  of  a  line  40,000 
feet  long  ;  what  will  be  the  increase  in  length? 


By   (38)  the   increase   is 
increased  length  40,200  feet. 


40,000 


=  200  feet,   giving  for  the 


B.  Locating  Simple  Curves. 

94.  To  Locate   a  Curve  with  the  Chain  by  Offsets   from 
Chords  Produced. 
In  Fig.  21  let  the  P.  C.  fall  at  B.     If  BC  is  a  full  chain,  prolong 


the  tangent  AB  to  //,  making  BII=  BC;  HG  will  equal  t,  which 
may  be  calculated  by  (32')  or  (33').  With  B  as  center,  strike  an 
are  with  radius  BIT,  and  with  II  as  center  and  t  as  radius  strike 
an  arc  ;  at  G,  where  these  arcs  intersect,  set  a  stake.  Produce 
BC  to  K,  making  CK  -  BC  =  CD  ;  strike  the  arc  KD  from  C  as 
center ;  make  the  chord  KD  —  d,  calculated  from  (29'),  (30'),  or 
(31).  Set  a  stake  at  D  and  proceed  in  like  manner  for  the  other 
points  until  the  P.T.  is  reached,  where  FPis  made  equal  to  t. 

Usually  the  P.O.  does  not  fall  at  a  full  station  ;  then  EC  =  t', 
which  may  be  found  by  (34)  or  (35).  Using  this  value  of  t',  we 
locate  C  as  above.  At  B  make  B R  •=  t',  and  prolong  RC  to 
L  ;  make  LD  =  t  and  set  a  stake  at  D.  EM  will  equal  d,  and 
may  be  located  as  before. 

We  may  regard  KD  as  equal  to  KL  -f-  t,  and,  finding,  KL, 


56        A   FIFLD-MANUAL   FOR  RAILROAD   ENGINEERS. 

measure  KD  and  set  D  without  locating  H.     To  do  this  we  have 
the  similar  triangles  BEG  and  CKL,  from  which 

KL  _    t' 
~CK~lBG' 

and  therefore,  since  KG  —  CD, 

—I- 

In  like  manner  at  Five  have 

TfTf 

PN=t^,    and    FP=V 
hence 


Make  EQ  =  </,  prolong  QF,  and  we  have  the  tangent  at  F. 
EXAMPLE.  —  Given  the  P.  (7.  of  a  5°  curve  at  106  +  20  and  the 
angle  of  intersection  22°,  to  locate  the  curve. 

oo 

Here  L  =  —  =  4.4  stations. 

o 

Therefore  the  number  of  the  P.  T.  is 

106.20  +  4.4  =  sta.  110  +  60. 
BG  in  this  case  is  80  ft.,  and  by  (33') 

t  =  0.873  X  5  =  4.37  ft. 


By  (35),         t'  =  4.37  X  =  2.80  ft. 

Set  off  HG  =  2.80  ft.,  and  at  D  make 

100 
KD  =  2.80  X  ~  r  +  4.37  =  7.87  ft. 

oU 

At  E  make  ME  =  d  =  8.72  by  (31).  This  will  be  at  sta.  109  ; 
at  110  set  a  stake  by  offsetting  8.72  ft.  The  last  chord  is  60  long, 
and  hence  the  offset 


NF=  4.37  X         +  4.37  X    —=  2.62  +  1.57  =  4.19  ft. 
Make  EQ  =  1.57  ft.,  and  prolong  QF,  the  terminal  tangent. 


LOCATION. 


57 


95.  To  Locate  a  D  Degree  Curve  by  Offsets  from  Tangent. 

Let  AM,  Fig.  22,  be  tangent  at  A,  and  E,  F,  O,  etc.,  points  on 
the  curve.     The  offsets  BE,   CF,    A  B 

etc.,  may  be  found  from  formula 
(36), 

z  =  lri*D, 

either  by  taking  equal  intervals, 
AB,  BO,  CM  along  the  tangent  or 
by  taking  E,  F,  G,  etc.,  at  regular 
stations  around  the  curve  and 
using  the  arc  length  instead  of 
the  tangent. 

When  the  arc  AG  is  large,  or 
strict  accuracy  is  required,  we 
proceed  to  find  the  offsets  at 
regular  stations  and  the  lengths 
of  AB,  AG,  etc.  First  find  R 
from  (12)  or  (12');  then  from  triangle  OEL, 


FIG.  22. 


BE  —  AL  —  R(l  —  cos  D]  =  R  vers  D, 
AB  =  LE  =  R  sin  D. 
In  like  manner 

CF  =  AH  =  R(l  -  cos  2D)  =  R  vers  2D, 
AC  =  IIF  =  R  sin  2D, 

and  so  on  for  any  number  of  stations. 

Should  .4  fall  at  a  plus  station,  we  first  find  the  angle  A  at  the 
center,  then 

BE  =  R  vers  D,  , 

AB  =  R  sin  £>, , 

CF  =  R  vers  (Z>,  -f  D), 

AC=  R  sin  (Z>i  -f  D\ 

etc.  =  etc. 

The  ordinates  BE,  CF,  etc.,  are  evidently  equal  to  the  mid- 
ordinates  for  long  chords  2LE,  2IIF,  etc.;  hence  we  can,  if 
A,  E,  F,  and  G,  fall  at  full  stations,  take  them  direct  from 
Table  V;  then  take  the  long  chords  from  Table  IV  and  dividing 
these  by  2,  get  the  required  coordinates. 


58      A  FIELD-MANUAL  FOR  RAILROAD  ENGINEERS. 

EXAMPLE. — Locale  three  stations  of  a  4°  curve  by  offsets  every 
50  ft.  on  curve. 

Referring  to  Table  V,  the  required  offsets  arc  0.87,  3.49,  7.85, 
13.94,  21.77,  and  31.31.  By  Table  IV  the  distances  measured 
along  tangent  are  50.0,  99.94,  149.76,  199.39,  248.78,  and  297.87. 
With  these  values  we  can  set  out  the  curve  either  way  from  A. 

Had  we  used  formula  (36)  we  should  have  had  for  the  values 
of  the  offsets  0.87,  3.50,  7.88,  14.00,  21.87,  and  31.50. 

96.  To  Locate  a  Curve  by  Offsets  from  a  given  Long 
Chord. 


Let  FK,  Fig.  23,  be  the  given  chord.     We  may  compute  the 
offsets  yi ,  y^ . .  .  M  by  the  methods  of  83— of  which  formula  (17), 

y  —  \rnnJ), 

is  the  most  convenient,   within  the  limits  of  its  applicability — 
and  setting  off  these  ordinates,  locate  the  curve. 

Or  we  may  set  off  the  rnid-ordinate  M  =  R  Trers  FOA  at  A, 
and  at  C  set  off  #2  —  M  —  JR  vers  D,  making 

AC  =  IIL  =  R  sin  D. 
GE  will  be 

y,  —  N  —  R  vers  2D,     and    AE  =  R  sin  27). 

ANOTHER  METHOD  is  1o  find  ilic  .-ingle  T\OFn.\,  the  center,  and 
by   Table  IX  determine  BA  —  M  ;  then  by  Tables  V   and  IV 


LOCATION".  50 

determine  BL,  BN,  LH,  and  NO.     Then  HO  =  M  -  BL,  which 
set  off  at  G,  and  other  points  in  like  manner. 

EXAMPLE.  —  Given  the  P.O.  of  a  4°  curve  at  station  160  -j-  75, 
the  angle  between  tangent  and  chord  =  9°,  required  the  offsets 
necessary  to  locate  the  curve. 

Here  7=2x9  =  18°. 

1  8 

.-.     L  —  —  =  4.50  stations. 
4 

Hence  the  P.T.  falls  at  160.75  -f  4.50  —  sta.  165  -j-  25.  The 
mid-point  on  curve  B  falls  at  sta.  163.  By  Table  IX, 

J,=  ™=  17.64  ft. 


By  Table  V  the  mid-ordinate  for  two  stations  of  a  4°  curve  is 

BL  =  3.49. 

Hence  I1C  =  17.64  -  3.49  =  14.15. 

By  Table  IV,          HL  =  AC  =  99.94  ft. 

Measure  AC  =  99.94  ft.,  and  set  off  CR  —  14.15  ft.,  and  drive  a 
stake  at  II.  In  like  manner  find 

<3#=3.70    and     .4  tf=  199.39  ft. 

The  points  P  and  Q  are  also  located  by  means  of  the  coordi- 
nates just  determined. 

If  B  had  fallen  at  an  odd  station,  the  curve  could  have  been 
located  in  the  same  manner,  Hand  P  being  100ft.  from  B,  G  and 
q  200,  etc. 

97.  To  Locate  a  Curve  with  Transit  and  Chain  when  the 
Degree  D  or  Radius  72  is  Known. 

If  R  is  given,  determine  D  by  (13);  then,  since  .he  angle  in 
the  circumference  of  a  circle  is  half  the  angle  at  the  center  sub- 
tended by  the  same  chord,  we  may  locate  points  on  the  curve  by 
successive  deflections  from  the  tangent. 

In  Fig.  24  let  the  P.O.  be  at  A,  at  which  point  set  the  transit, 
and  with  the  vernier-plates  clamped  at  zero  place  the  telescope 
in  tangent  either  by  sighting  the  P.I.  or  by  backsightiug  to  some 
point  in  the  tangent  Deflect  from  the  tangent  half  the  angle  at 
the  center  for  the  sub-chord  or  chord,  and  direct  the  head  chain- 
man  into  line  while  the  rear  chainman  holds  his  end  of  the  chain 


GO        A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

at  the  transit,  the  chain  being  kept  taut.  The  stakemaii  drives  a 
stake  at  the  point  where  the  head  chainman's  flag  rested,  and  the 
rear  chainman  advances  to  this  point.  Deflect  %D  from  the  chord 
AB  just  run,  and  while  the  rear  chaiuman  holds  his  end  of  the 
chain  at  B  direct  the  head  chainman  into  line  at  C.  Other  points 
are  located  by  deflecting  an  additional  \D  for  each  chord  length 
measured,  until  a  point  E  is  reached  to  which  it  is  desirable  to 


Fio.  24. 

move  the  transit.  The  angle  FAE  should  not  exceed  about  15°. 
Move  the  transit  to  E,  backsight  to  A,  and  deflect  FEA  =  EAF, 
when  the  telescope  will  be  in  tangent,  and  the  curve  can  be  con- 
tinued until  it  is  again  necessary  to  move  the  transit.  At  the 
P.  T.  put  the  telescope  in  tangent  by  backsighting  to  the  point 
last  occupied  by  transit  and  deflecting  the  tangential  angle  as  at 
E.  The  line  may  now  be  continued. 

98.  The  Index-angle  is  read  on  the  vernier-plate,  and  is  the 
angle  between  the  tangent  to  the  curve  at  the  P.  C.  and  any  other 
line  passing  through  a  point  on  the  curve  when  the  telescope  is 
directed  along  this  line.     It  is  most  frequently  taken  as  the  angle 
between  the  initial   and   any  subsequent  tangent  to  the  curve. 
Thus  at  E  the  index-angle  equals  EFP  =  2FAE.     At  any  point 
on  the  curve  the  index-reading  in  tangent  may  be  found  by  the 
following  rule,  which  may  be  easily  deduced  from  a  figure: 

From  double  the  index-angle  that  fixed  the  point  subtract  the  index- 
angle  in  tangent  at  the  last  point;  the  remainder  is  the  index-angle 
required, 

99.  Subdeflection-angles   may  be  found   by  (13)   rigidly,  or 
approximately  (and  with  sufficient  accuracy  except  when  D  is  very 
large)  by  assuming  the  central  angles  to  be  proportional  to  their 
chords.     Thus  on  a  4°  curve  the  central  angle  for  a  sub  chord  of 
25  ft.  would  be  1°,  and  the  subdollertion -angle  30'. 


LOCATION.  61 

EXAMPLE.— Locate  a  4°  curve  to  left  when  the  P.O.  is  at  sta. 
81  -I-  25  and  /=  32°  36'. 

Here  L  -  ^  =  8.15  chains. 

Hence  the  P.  T.  will  fall  at  81.25  +  8.15  =  sta.  89  -f-  40.  The 
first  sub-chord  is  75  ft.  long,  and  the  first  deflection-angle  will  be 
found  by  (12). 

nfj    K 

SiD^  =  1-432.7  =  °-02617 

.'.     £<^=-l°    30'. 

By  the  approximate  rule,  since  |D  =  2°, 

id  _  75 
~2    ~I66' 

whence  |S  =  2  X  I  =  1°  30'  as  before. 

With  transit  at  P.O.  deflect  1°  30'  from  tangent,  measure  75 
feet,  and  set  sta.  82.  Then  a  deflection  of  3°  30'  will  determine 
83,  5°  30'  sta.  84,  7°  30'  sta.  85.  Now  remove  transit  to  85,  and 
with  vernier  at  7°  30'  backsight  to  81  +  25.  Reverse  telescope 
and  set  vernier  at  15°  00',  when  the  telescope  will  be  in  tangent. 
An  index  angle  of  17°  will  fix  86,  and  so  on. 

The  last  chord  will  be  oiity  40  feet  long,  for  which  the  sub- 
deflection-angle  is  T4(fy  of  2°,  that  is,  48'.  The  index-angle  fixing 
the  P.T.  is  therefore  23°  48'. 

To  get  in  tangent  at  89  -f  40  backsight  to  sta.  85,  with  vernier 
at  23°  48' ;  then  by  the  rule  of  98  the  index-reading  is  (23°  48')  X 
2  —  15°  =  32°  36'  =  /.  Set  the  vernier  at  this  reading  and  run 
tangent. 

CAUTION.  — It  is  not  good  practice  to  set  more  than  4  or  5  sta- 
tions on  curve  from  any  one  point.  MR.  SHUNK  gives  the  limit- 
ing angle  to  be  deflected  from  tangent  as  20°,  and  says  15°  should 
rarely  be  exceeded.  (Field  Engineer,  p.  82.) 

100.  The  Transit  Notes  may  be  conveniently  kept  in  the  form 
below,  which  shows  the  notes  for  the  last  example. 

When  possible  the  tangents  should  be  run  to  intersection,  the 
angle  1  measured,  and  the  tangent  distance  calculated.  Then 


62        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS 


a 
•2® 

,  be 

c  « 
'2  S 

Is 

*t 

Station. 

%Jp 
|« 

11 

£be 

11 

11 

O 

II 

Remarks. 

90 

-HO 

QP.T. 

0°48' 

23°  48' 

32°  36' 

N27°36'E 

N  27°30'  E 

89 

23°  0' 

88 

21°   0' 

87 

19°  0' 

86 

17°  0' 

85 

O 

7°  30' 

15°  0' 

84 

5°  30' 

83 

2°    0' 

3°  30' 

82 

1°30' 

1°30' 

4°  C.L.;  P.L  set. 

+25 

0P.C.4°C.L. 

0°   0' 

0°  0' 

0°  0' 

/  =  32°  36';     T  - 

418.9  ft. 

81 

N  60°12'  E 

N  60°10'  E 

measure  along  tangents  and  set  P. C.  and  P.T.  from  the  P.L 
When  the  curve  is  run  iu;  the  position  of  the  P.T.  thus  found 
should  agree  with  the  one  set  from  the  P.I.  If  the  error  is 
greater  than  the  circumstances  of  the  case  permit,  the  curve 
must  be  rerun  and  tangents  remeasured. 

101.  Another  Form  of  Notes,  and  in  some  respects  a  better  one 
than  the  above,  is  given  below.  The  index-readings  are  com- 
puted as  though  the  entire  curve  were  run  from  the  P.  C.  The 
notes  for  the  last  example  would  appear  as  below  : 


§<u 

*l? 

—  o5 

•d 

-  w' 

fj 

Station. 

o"5b 

^^ 

~  S 

|| 

o  o 

11 

Remarks. 

r 

H4  y 

5° 

IS 

90 

4-40 

QP.T. 

0°48' 

16°  18' 

32°  36' 

N  27°36'  E 

N  27°30'  E 

89 

15°  30' 

88 

13°  30' 

87 

11°30' 

86 

9°  30' 

85 

O 

7°  30' 

84 

5°3C' 

83 

2°   0' 

3°  30' 

82 

1°30' 

1°30' 

4°  curve  left  ; 

4-25 

0P.CU°C.L. 

0°    0' 

0°  0' 

P.I.  set.  7=32°16'; 
2'  =418.9  ft. 

81 

N60°12'E 

N60°10'N 

The  computations  are  all  made  before  beginning  the  work,  and 
the  notes  have  the  advantage  of  permitting  the  tracing  of  the 
curve  either  way  from  the  instrument  without  additional  compu- 


LOCATION.  63 

tations.  Suppose  the  trausitman  to  have  rim  the  curve  from  the 
P.  C.  to  sta.  85,  to  which  point  he  removes  the  instrument.  He 
there  sets  the  vernier  at  0° — the  angle  on  limb  when  telescope 
was  iu  tangent  at  the  P.  C. — then  sighting  the  P.  C.  he  reverses 
the  telescope  and  deflects  to  9°  30',  which  will  fix  sta.  86.  Had 
the  tangent  at  85  been  desired,  a  reading  of  7°  30' — the  angle  that 
located  that  point — would  have  put  the  telescope  in  the  plane  de- 
sired. A  reading  of  11°  30'  fixes  87,  and  so  on  to  the  P.  T. 
Removing  to  the  P.T.,  the  plates  are  clamped  at  7°  30',  and  a 
backsight  to  sta.  85  taken ;  then  deflecting  to  16°  18',  the  tele- 
scope is  in  tangent  at  the  P.T.  Had  it  been  desirable  to  set  84 
from  85,  a  reading  of  5°  30'  would  fix  that  point ;  others  may 
be  found  in  the  same  manner. 

Any  convenient  form  of  notes,  which  are  intelligible  to  another 
engineer  who  may  have  to  retrace  the  curve,  may  be  used,  but  it 
is  desirable  that  some  general  form  should  be  employed.  Either 
of  the  preceding  forms  seems  to  meet  ordinary  requirements. 

C.  Obstacles. 

102.  To  Pass  an  Obstacle  on  a  Curve. 

FIRST.  Suppose  the  obstacle  to  be  one  obstructing  vision  at  one 
station  only. 

In  Fig.  25  suppose  transit  set  at  A,  and  B  and  C  located  from 
that  point,  but  the  next  full  station,  II,  to  be  invisible  from  A. 


FIG.  25. 


Set  a  plus  station  at  E,  as  near  the  obstruction  as  may  be  conven- 
ient, then  set  F 100  feet  from  E.  Next  make  FG  =  100  -  GE, 
and  locate  O  with  the  corresponding  deflection-angle.  Other 
stakes  may  be  set  beyond  G,  or  the  transit  may  be  removed  to 
that  point  and  the  curve  beyond  traced. 

SECOND.     Suppose  the  line  of  siylit  obscured  for  more  than  one 
station,  as  in  Ft'y,  20. 


64        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


If  transit  is  at  A,  deflect  an  angle  HAB  that  will  clear  all  ob 
structious,  aud  at  the  same  time  cause  B  to  fall  at  a  full  station. 
Then  by  Table  IV,  Table  IX,  or  by  formula  (16)  calculate  the 
long  chord  A13  ;  measure  AB  and  move  transit  to  B  ;  then  dellect 


FIG.  26, 

the  angle  ABC=  BAH  when  the  telescope  will  be  in  tangent. 
The  curve  may  now  be  run  both  ways  from  B. 

If  it  happen  that  some  stations,  as  E  and  F  in  the  figure,  are 
still  invisible,  they  may  be  located  by  offsets  from  chord  or  tan- 
gent. 

EXAMPLE. — Let  the  curve  be  a  3°  curve  to  right  -,  angle  NAB 
=  7°  30',  the  deflection-angle  for  5  stations.  By  Table  IV  the 
long  chord  is  498.63  feet,  which  can  now  be  measured  and  a  hub 
set  at  B ;  then  making  angle  CBA  =  7°  30',  the  telescope  will  be 
in  tangent  and  the  curve  can  be  traced  either  way. 

103.  To  Locate  a  Curve  when  the  P.O.  is  Inaccessible. 

In  Fig.  27  let  the  P.  C.  at  B  be  in- 
accessible ;  it,  is  desired  to  reach  a 
point  II  on  accessible  ground. 

FIRST  METHOD.  —  Assume  a 
point// on  the  curve  such  that  a 
line  AH  from  an  accessible  point 
A,  on  tangent,  will  clear  the  ob- 
stacle ;  for  convenience  H  should 
be  at  a  full  station.  The  arc  BI1 
and  central  angle,  which  equals 
HCF,  are  then  known.  Calculate 
BC  =  T  by  (14)  or  (Ua)  ;  then 
since  AB  is  known,  AC,  =  AB -f- 
BC,  is  known. 

Now  in  triangle  A  CH,  from  trig- 
onometry, 

tan  \(h  -  a)  _  A  G-CH 
tan  l(7i  +  a) 


FIG.  27. 


LOCATION.  65 

But  (h  -f-  a)  =  c  •  hence 

A  r1     rtj 

tau«A-a)  =  -^TWtan^  .....  (40> 

Then  \(h  -J-  a)  -f-  |(7i  —  a)  =  h,  the  larger  angle,  and 
|(A  -}-«)  —  |(7i  —  «)  =  a,  the  smaller  angle.  AH  may  be 
found  by  the  law  of  sines,  or  by  drawing  CE  perpendicular 
to  AH,  when 

AH  =  AC  cos  a  +  CHcosh  .....     (41) 

EXAMPLE.—  The  P.  C.  of  a  4°  curve  is  at  sta.  141  -f  25,  and  it 
is  desired  to  reach  the  point  J2"from  sla.  139  on  tangent. 

Suppose  H  be  assumed  to  fall  at  sta.  147  ;  the  curve  length  is 
L  =  147  —  141.25  =  5.75  chains.  Then  angle  c  =  5.75  X  4  = 
23°  0'.  By  Table  IX  the  tangent  distance  for  a  1°  curve  is 
Ti  4  23°  =  1165.8  ft. 


By  (14«),  T  =  =  291.45  ft. 

Now  AC  =  291.45  -f-  225  =  516.45  ft., 

and 

AC+  CH=  516.45  +  291.45  =  807.90, 
while 

AC  -  CH  =  225  ft.  ; 
hence,  by  (40), 

2^)F) 

tan  l(h  -  a)  =  —  —  „-  X  0.20345  =  0.05666  =  tan  3°  15'. 
807.9 

Therefore 

7i  =  11°  30'  +  3°  15'  =  14°  45', 
and 

a  =  11°  30'  -  3°  15'  =    8°  15'. 
By  (41), 

AH=  516.45  x  0.98965  +  291.45  X  0.96705  =  793.0  ft. 

At  A  deflect  8°  15'  from  tangent,  measure  793.0  ft.  and  set  a 
hub  ;  move  to  this  point,  backsight  to  A  and  deflect  14°  45'  into 
tangent,  then  trace  in  the  curve. 


G6        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

SECOND  METHOD. — If  F,  any  assumed  point  in  tangent,  is 
visible  from  A,  AF  may  be  measured  by  some  indirect  method; 
then  AF —  AB  =  T.  The  tangent  for  a  1°  curve  having  same 
intersection-angle,  KFG,  is  7\  =  Tx  If;  find  this  value  of  Ti  in 
Table  IX  and  take  out  the  corresponding  value  of  L  "With 
transit  at  F  deflect  the  angle  KFG,  measure  FG  =  FB  =  T,  and 
set  hub  at  G.  The  station  number  of  G  is  found  by  dividing  the 
central  angle,  =  KFG,  by  the  degree  of  curve  D.  Move  to  G  and 
trace  the  curve. 

EXAMPLE.— Let  AF  measure  490.5  ft.  from  sta.  139  of  the  last 
example.  Then  AB  =  225  ft.,  and  BF=  490.5  -  225  -.=  265.5  ft. 
265.5  X  4  =  1062  ft.,  which  by  Table  IX  is  the  value  of  2\  for 
/=  21°.  Set  transit  at  F,  deflect  21°,  and  measure  FG  =  265.5  ft. 


21 
L  =  —=5.25  chains; 


hence  G  will  fall  at  141.25  -f  5.25  =  sta.  146  +  50.  Move  to  G 
and  run  the  curve  both  ways. 

THIRD  METHOD. — In  Fig.  28  let  the  inaccessible  P.  C.  be  at  B, 
and  let  it  be  required  to  reach  E  from  a  point  G  on  the  curve 
prolonged  backwards  from  B. 

At  a  given  point  A  on  tangent  cal- 
culate the  tangent  offset  by  (36)  or 
the  methods  of  95,  then  set  this  off  at 
right  angles  to  AB  ;  set  the  transit  at 
C  and  turn  off  ACL  =  90°  -  COB, 
when  the  telescope  will  be  in  tangent 
at  C.  COB  may  be  found  from  Table 
IX  by  multiplying  AC  by  the  degree 
of  curve  and  taking  half  the  intersec- 
tion-angle corresponding  to  the  mid- 

ordiuate  that  equals  this  product.  Now  deflect  and  measure 
ECL,  then  by  (16)  or  (16«)  calculate  GE,  which  measure.  Move 
to  E  and  deflect  LEG  =  ECL  and  the  telescope  will  be  in 
tangent.  The  central  angle  BOE  =  2LEC  -  BOG,  from  which 
the  arc  BE  &ud  number  of  sta.  E  may  be  found. 

EXAMPLE. — Take  the  same  example  as  in  the  last  two  cases. 
A  is  at  sta.  139,  B  at  141  -f  25;  hence  AB  =  2.25  stations. 


By  (36),        z  =  AC=  •  X  (3.35)»  X  4  =  17.72  ft. 


LOCATION. 


67 


Or  by  Table  IX  the  angle  corresponding  to  the  long  chord 
(2  X  2.25)  X  4  —  1800  ft.  is  18°  4',  for  which  the  mid-ordinate  is 

71.06  ft.     For  our  4°  curve  the  mid-ordiuate  will  be  — '-—  •=.  17.77 

4 

ft.,  which  equals  AC  and  agrees  closely  enough  with  the  value 
for  z  above. 

Make  angle  BAG  =  9Q\  and  measure  AC  =  17.72  ft.  Move 
to  C  and  sight  to  A,  then  make  angle  ACL  =  90°  —  (9°  2')  = 
80°  58'.  Suppose  an  angle  LCE  =  16°  1'  to  clear  the  obstacle. 
By  formula  (16), 

CE  =  2R  sin  (16°  1')  =  2  X  1432.7  X  0.27592  =  790.6  ft. 

Measure  along  CE  790.6  ft.  and  set  a  hub;  move  to  E  and  run 
the  curve. 

CE  might  have  been  found  by  means  of  Table  IX,  for  the  long 
chord  of  a  1°  curve  having  1  =  2LCE  =  32°  2'  is  3162.0  ft.; 
divide  this  by  4  and  there  results  CE  =  790.5  ft. 

104.  To  Pass  to  Tangent  when  the  P.T.  is  Inaccessible. 

This  is  just  the  reverse  of  the  preceding  problem,  and  may  be 
accomplished  by  reversing  the  processes  described  above. 

When  the  P.T.,  however,  falls  in  or  beyond  a  river  or  lake 
obstructing  the  ordinary  methods  of  indirect  measurement,  the 
ease  merits  a  special  solution. 

FIRST  METHOD.— In  Fig.  29  let  the  transit  be  at  A,  and  B  the 
P.T.      From    the    known   station 
numbers  of  A  and  B  the  length  of 
curve  and  angle  /  may  be  found; 
then,  by  (14),  AC  =  11  tan  |/,  or,  V  \  ,r 


by(14a),  AC=. 

Move  to  C  and  deflect  the  angle 
/;  set  a  stake  F,  and  one  at  some 
other  accessible  point  E\  measure 
angle  ECF  '=  c.  Move  to  F  and 
measure  the  angle  EFC  and  the 
side  EF;  then  in  triangle  ECF 
angle  e  =  180°  —  (c  -(-/);  by  trigo- 
nometry 


FIG.  29. 


. 
sm  c 


(42) 


68        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


Since  BC  =  AC,  there  results  BF  =  CF  —  AC;  and  as  the  sta- 
tion number  ut  B  is  known,  that  at  F  becomes  known,  and  the  liue 
may  be  continued. 

If  B  is  not  the  P.T.,  measure  back  the  distance  FB,  set  transit 
at  B,  and  continue  the  curve. 

EXAMPLE.—  Let  the  P.T.  of  a  2°  C.L.  fall  at  sta.  205  -f  50—  an 
inaccessible  point;  suppose  A  at  sta.  200,  angle  c  =  40°,  /=  80°, 
JSF=310ft. 


Here 


',     and      e  =  60°. 


T  = 


=_  375.37  ft. 


From  (42),  applying  logarithms, 


log  CF=  2.49136  -f  9.93753  -  9.80807  =  2.6082. 


Whence  CF=  417.7  ft.     Then  BF—  417.7  -  275.87  -  141.8  ft.; 

therefore  the  number  of  F  \vi\l  be  206  +  91.8. 

SECOND  METHOD.  —  In  Fig.  30,  with  the  transit  at  any  point  A 
on  the  curve,  assume  a  long  chord  AB 
and  calculate  the  angle  CAB;  deflect 
this  angle  from  the  tangent  AC,  and  set 
a  point  E  beyond  obstruction  ;  set  also 
a  stake  at  C  in  tangent. 

Move  to  E  and  measure  A  EC  and 
side  EC.  Compute  AE  from  the  trian- 
gle AEG.  If  this  is  greater  or  less 
than  the  length  of  the  long  chord  AB, 
take  their  difference  BE  and  set  a  hub 
at  B.  With  the  transit  at  B  trace  out 
the  curve. 

EXAMPLE.  —  Given  A  at  sta.  210  of  a 
3°  C.  L.,  angle  a  =  12°,  b  =  92°,  EC 
=  181  ft.  Then  c  =  76°,  and  by  solving 

the  triangle  AEG,  AE-  844.7  ft.    By  Table  IX  the  long  chord  of 

OOQO  ft 

a  1°  curve  for  /=  24°  is  2382.6  ft.  ;  therefore  AB  =         ' 


FIG.  30. 


o 

ft.     Now  will  .##  =  844.7  —  794.2  =  60.5  ft,,  which  is 
tance  along  EA  that  transit  must  be  moved  back  from  E. 


=  794.2 
the  dis- 


LOCATION. 


69 


105.  Given  the  Perpendicular  p  from  a  Point  to  a  Tangent, 
to  Find  the  Point  on  Tangent  at  which  to  Begin  a  Curve  of 
Given  Radius  which  will  Pass  through  the  Given  Point. 

FIRST  SOLUTION. — In  Fig.  31  let  P  be  the  point,  BP  the  per- 
pendicular.     We    have    to    find          , 
BA  =  x.  A[< 

From  P  draw  PC  parallel  to 
AB ;  then  in  triangle  OPG 

K*  =  a?  +  CR  -  p)*. 
From  which 


x  =  \/2Rp  -  p\     .     (43) 

SECOND  SOLUTION. — Consider 
p  —  AC  as  the  mid-ordiuate  for 
a  long  chord  =  2x  ;  then  p  X  D 
=  the  mid-ordinate  for  a  1°  curve 
for  a  central  angle  equal  2a. 
The  corresponding  long  chord  may  be  taken  from  Table  IX. 
Then 


FIG.  31. 


IL.C. 


(43«) 


EXAMPLE.— Given  p  =  30  ft.,  D  =  4°  (E  =  1432.7),  to  find  x. 


By  (43),         x  =  V85.962  -  900  =  291.65  feet. 
By  the  second  method, 

30  X  4  =  120, 

the  mid-ordinate  for  a  1°  curve  corresponding  to  an  angle  of 
23°  29',  for  which  the  long  chord  is  2332.6.     Now,  by  (43a), 


=  291.6  feet. 


106.  In  Fig.   31,  Given  x  and  p  to  Find  the  Radius   of  a 
Curve  Tangent  to  AB  at  A  and  Passing  through  P. 


From  (43), 


2P 


(44) 


70        A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

107.  Given  the  Location  of  a  Point  P  referred  to  the  P.  1. 
to  Find  the  Radius  of  a  Curve  through  P  which  will  Unite 
the  Given  Tangents. 


B 


FIG.  32. 


In  Fig.  32  suppose  BC  —  I,  BP  =  m  known,  and  angle  a  cal- 
culated ;  or  PC  and  a  may  be  measured  on  the  field. 
From  triangle  CAO, 


b  =  90°  -  (a  + 
Now  from  triangle  PCO, 


and     CO  =  R  sec 


CO    . 

sm  y  =  po  sm 


Inserting  values  of  PO  and  CO, 

R  sec  \I 

y  —  -          .  sin  b  =  sec  \l  .  sin  5  = 


sin 


sin  b 

-  —  , 


an  equation  from  which  the  unknown  11  has  disappeared. 
from  the  same  triangle,  since  x  =  180°  —  (b  -f-  y}, 


sm 


When  I  =  90°,  it  can  easily  be  shown  that 


•     (45) 

Next, 

.     (46) 
(47) 


LOCATION.  71 

108.  To   Locate   a  Tangent   to    a    Curve   from   an  Outside 
Point. 
FIRST  METHOD.— In  Fig.  33  let  P  be  the  point  and  AHB  the 


.,    R/ 


/o 

Fia.  33. 

curve.  Run  a  trial-line  PA  cutting  the  curve  in  A  and  B. 
Measure  PA  and  AB ;  or  measure  PA  and  angle  a  between  the 
chord  AB  and  tangent  AL.  Then 

AB  —  2AC  =  2R  sin  a, 
00  =  R  cos  a. 

By  geometry,  PE  =  VPA  X  PB,  PE  being  the  required  tan- 
gent.    From  the  figure, 

CO 


tan  m  = 


_ 
PE' 


At  P  deflect  the  angle  I  =  m—  n  from  PA  and  run  the  tangent. 
SECOND  METHOD. — In  Table  IX  find  the  long  chord  for  a 
central  angle  2a  ;  then 


AB  =  2A  C  = 


L.O. 


~D> 

and  CO  =  R  —  CH. 

"We  may  now  proceed  as  before. 


72        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

109.  To  Run  a  Tangent  to  Two  Located  Curves  of  Contrary 
Flexure. 

FIRST  CASE.— In  Fig.  34  let  FK  and  LE  be  the  curves,  and 
KL  —  p  measured  on  the  ground. 


FIG.  34. 


Let  F1S—  t  be  the  required  tangent. 

Draw  OiH  parallel  and  0*H  perpendicular  ioFE;  from  the 
triangle  OJIO*  ,  since  FH  =  R,  , 


(JB, 


whence 


t= 


.    .     .     .     (48) 


Also, 


cos  a  — 


(49) 


The  arcs  FK  and  LE  may  be  found  from  the  angle  a  and  the 
known  curvatures,  after  which  the  points  .F  and  .E'may  be  set. 

If  t  is  given  and  p  required,  it  may  easily  be  found  from  (48). 

SECOND  CASE,    p  not  known. 

Set  the  transit  at  a  point  A  on  one  curve. and  note  the  bearing 
of  tbe  tangent  to  the  curve  at  that  point  (see  Fig.  34);  the  bearing 
of  the  radius  0?A  differs  from  this  by  90°.  Run  a  line  ABC  of 
one  or  more  courses  to  intersect  the  oilier  curve  at  C.  Note  the 
bearings  and  lengths  of  these  courses  and  the  bearing  in  tangent 
at  C,  from  which  calculate  the  bearing  of  CO^  Rl  and  7?2  being 
known,  the  latitudes  and  departures  are  next  calculated.  Let  02A 


LOCATION. 


73 


be  the  sum  of  the  northings  or  southings,  01^  the  sum  of  the 
eastings  or  westings  ;  from  the  triangle  0i  022V, 

tan  b  =  -=^-T, 


and 


As  before,  FE  is  the  required  tangent  and  02£T  perpendicular, 
while  OiH is  parallel  thereto. 


coa  a  — 


0,0, 


Angle  F0itf  =  b  -  a  is  the  bearing  of  0*F,  while  AO*F  = 
c  —  b -\-  a  is  the  angle  of  retreat  from  the  known  point  A  to  F, 
where  the  tangent  may  be  run.  The  length  of  t  = 

t  =  00,  sin  a. 


D.   Change  of  Location. 

110.  To  Locate  a  Curve  Parallel  to  a  Given  Curve. 

Let  p  be  the  perpendicular  between  parallel  tangents,  and  sup- 
pose ABC  located  (see  Fig.  35). 
If  there  are  no  restrictions  as  to  the 
position  of  the  points  E,  F,  and  Q 
on  the  second  curve,  we  may  cal- 
culate the  new  degree  of  curve  Di 
for  a  radius  Ri  =  R  -f  p,  by  (13), 
and  trace  the  curve  from  any 
point,  as  E.  Thus 


50 


50 


If,  however,  points  on  the  radii 
through  A,  B,  and  G  are  wanted, 

they  are  gotten  by  using  the  same  degree  of  curve  D  and  com- 
puting the  length  of  chord  FE.     From  similar  triangles, 


EF 


AB 
R 


100 


74        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 
whence 


EF  =  100  ?!  =  100  ?4r^  =  100  (l  -f  |  ).  .         (50) 
ri  H  >          .a* 


Had  .SFY?  been  the  located  curve,  with  radius  R,  we  should 
have  had 


(51) 


111.  To  Change  the  P.O.  of  a  Located  Curve  so  that  P.T. 
will  Fall  in  a  Given  Tangent  Parallel  to  Terminal  Tangent  of 

Located  Curve. 

Let  AB,  Fig.  36,  be  the  lo- 
cated curve  ;  FE,  the  tangent 
in  which  the  P.I.  must  fall. 

Let  the  distance  between  tan- 
gents be  HE  =  p. 

Draw  BE  and  00'  parallel  to 
AF ;  evidently  AC  =  00' =BE, 
O'  being  the  new  position  of 
center. 


In  triangle  BEH, 


BE  =  AC  =  ~-j.  =  p  cosec  /. 


(52) 


Set  the  new  P.  C.  by  measurement  from  A,  and  run  the  curve 
CE.  Any  system  of  straight  lines  and  curves  may  be  treated  as 
above,  provided  1  is  the  angle  between  initial  and  terminal 
tangents  and  p  as  before. 

EXAMPLE. — A  located  2°  30'  curve,  having  /  =  25°,  ends  in  a 
tangent  25  ft.  outside  of  desired  tangent.  Find  the  change  in 
position  of  P.  C. 


By  (52), 


AC  =  25  X  2.36620  =  59.16  ft. 


LOCATION. 


75 


C     K 


112.  To  Find  the  Change  in  Radius  and  Position  of  P. G.  if 
P.T.  is  Required  to  fall  on  the  same  Radial  Line  but  on  a 
Tangent  distant  p  from,  and  parallel  to,  Terminal  Tangent  to 
Located  Curve. 

In  Fig.  37  let  AB  be  the  located  and  CE  the  required  curve. 
Draw  the  parallel  chords  AB  and 
GE.  Draw  CHaud  BF perpendicular    _A_ 
to  AB.  The  angles  FBE=  CAH—\If 

From  the  figure, 

CH—  AC  sin  |7, 
BF  —  BE  cos  |7  =  p  cos  |7. 
Equating,  p 

A  G  sin  \I  =  p  cos  ^7,  Q 

whence 


FIG.  37. 


(53) 


In  the  triangle  OPOt,  0,P  =  AC,  OP  -  R  -  7?,,  and 

(R  —  R,)  tan  /  =  AC  =  p  cot  \I, 
or  R  -  R,  =  AC  cot  /  =  p  cot  \I  '.  cot  /. 

Therefore 

7?,  =  R  —  AC  cot  I  =  R  —  p  cot  4/.  cot  7.  . 

From  trigonometry, 

* 


(54) 


coti/= 


1  —  cos  1 
Inserting  these  values  in  (54)  gives 


and    cot  7= 


—  H  —  p 


sin  7        cos  7 


cos  7 


cos  7 


.  -  --  -  .  —  - 
1  -  cos  7    sin  7 


From  trigonometry, 


ex  sec  7  = 


T  —   72  ll =. 

1  —  cos  7  *  vers  7 


vers  7 
cos  7* 


?!  =72- 


ex  sec  7' 


(54') 


EXAMPLE. — A  2°  30'  curve  strikes  25  ft.  inside  a  tangent  in 
which  the  P.  T.  must  fall.  Find  the  necessary  change  in  radius 
and  position  of  P.C.  when  /  =  25°. 

By  (53)  the  change  in  P.  C.  is 


By  (54'), 


AC  =  25  X  4.51071  =  112.77  ft. 
25 


,  =  2292.01  - 


=  2050.38  ft. 


.10338 
By  Table  I  we  find  this  to  be  the  radius  of  a  2°  47'  41"  curve. 

113.  Given    a  Located    Curve  uniting    Two   Tangents  to 
Find  the  Change  in  Position  of  P.  C.  or  in  Radius  for  a  Given 
Change  in  the  Intersection-angle. 
FIRST  CASE. — Radius  unchanged. 

In  Fig.  38  let  BCE  =  /be  the  origi- 
nal intersection-angle,  FCE  =  I'  the 
new  angle.     From  the  figure, 
AG  =  AC-  GC, 

AG  -  R  (tan  \I  ~  tan  \1'\    (55) 

BY  TABLE  IX.— From  the  table,  for 
angle  /, 


Then 

SECOND  CASE. — P.O.  unchanged. 

Here  the  tangent  T  for  the   two  curves  is  the  same,  and 
therefore 

Ri  tan^/'  =  .Rtan^/; 

(56) 


Whence 


i  =  Jltanf/.cotfr. 


LOCATION. 

BY  TABLE  IX, 

y,  41*     Ti4r° 
D  D, 


114.  To  Find  the  Change  in  R  and  P.  C.  for  a  Given  Change 
in  /,  the  P.T.  remaining  unchanged 


'O, 
FIG.  39. 
In  Fig.  39,  from  the  triangles  OBG  and  OiBH, 

OG  =  R  cos  / 
and  01H=RlcosI1. 

Now  GA  —  HF;  hence 

Ri  —  Ri  cos  /i  =  R  —  R  cos  /. 
Whence 

...         (57) 


vers  li 

Also,  FA  =  HG  =  BH  -  BG. 

Inserting  values  of  BH  and  BG,  there  results 

FA  =  R,  sin  I,  -  R  sin  /. (58) 

115.  Given  a  Located  Curve  to  Find  the  Change  in  R  for 
a  Given  Change  in  T,  I  remaining  unchanged. 

In  Fig.  40,  from  the  triaugles  OAC  and  0,EC}  since 
EA  =  EC  -  AC, 


78        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


R!  tan  ±1  -  R  tan  |J  =  EA  =  T'  -  T. 
Whence  R,  =  R-\  -  (T1  -  -  T)  cot  {1 (59) 

_ .  E         A  c 


FIG.  40. 
BY  TABLE  IX.—  EA  being  known,  T7'  =  T+  EA-.    Then,  by 


If  the  change  in  vertex  of  curve  is  wanted,  there  results,  from 
(25), 

E  =  CO  =  T  tan  i/,       E'  =  GH  =  T'  tan  J/. 
Therefore         OH  =  E'  -  E  =  (T'  -  T)  tan  */,..,     (60) 

GHc&n  be  found  from  Table  IX  after  finding  Dl  as  above. 
If  Ri  is  given  and  EA  wanted,  (59)  yields 

EA  =  T'  -  T  =  (R1  -  R)  tan  %I. 

116.  To  Find  the  Radius  of  a  Curve  having  the  Same  P.G. 
as  a  Given  Curve,  but  ending  in 
a  Parallel  Tangent. 

In  Fig.  41  let  the  perpendicular 
distance  between  tangents  be  p,  and 
AB  be  the  located  curve;  A0t  =  R! 
is  required. 

FIRST  METHOD.  —  Draw  OH  at 
right  angles  to  OiE;  then 

OiE  =  dll  +  NO  +  OE, 
or 

Ri  =  (Ri  -  R)  cos  I  -  j-  7?  +  p. 


FIG.  41. 
From  which    Ri 


=  .R  + 


P         _ 
1  —  cos  / 


p 


vers  / ' 


(61) 


LOCATION.  79 

SECOND  METHOD.—  A,  B,  and  E  lie  on  the  same  straight  line, 
since  /  is  the  same  for  both  curves.  In  triangle  BOE  angle 
EBG  =  7,  and 


From  Table  IX,      AB  = 


AE  =  AB  +  BE  is  the  long  chord  for  curve  of  degree 
therefore 


If  desired,  JR  may  be  found  by  (12')  or  Table  I. 
THIRD  METHOD. — Draw  FL  parallel  to  0\E\  then 


CF  =  — — -  =  p  cosec  /. 
sm  /      L 


From  Table  IX,       AC  =  ~fj—. 

AF=  AC-}-  CF,  the  tangent  distance  for  second  curve  ;  hence 


D,  = 


AF 


REMARK. — If  transit  is  set  up  at  B,  it  will  be  well  to  set  E 
by  measurement  from  B,  to  serve  as  a  check  when  the  curve  is 
run  in  from  A. 


80        A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


ARTICLE  9.     COMPOUND  CURVES, 
A.  Location  Prob ferns. 

117.  Given  Two  Unequal  Tangents,  their  Intersection-angle, 
and  One  Radius,  to  Find  the  Other  Radius  of  a  Compound 
Curve  uniting  Tangents. 

In  Fig.  42,  AH  =  T!  and  BE  =  T2  are  the  known  tangents, 
AOi  =  Ei  the  known  radius.  BOi  =  R2  and  the  angles  /i  and  J2 
must  be  found  before  curve  can  be  located. 


Extend  first  branch  to  F,  so  that  tangent  FL  is  parallel  to  BIT. 

Draw  HK  and  BG  perpendicular  to  FL  ;  draw  FB  and  extend 
to  E\  it  will  pass  through  the  P.C.C.,  because  the  central  angles 
EO,F  and  EO*B  are  equal.  Then 

To  =  AL  =  Hi  tan  ^7. 
In  triangle  LHK,  since  LH  =  T0  —  T!  , 


p  =  HK=  BG  =  (T0-  T7,)  sin  Z 
Now  in  triangle  BGF  angle  BFG  =  i/a,  and 


LOCATION.  81 

tanifi  =  f  ......     ...     (62) 

Draw  0a  17  parallel  to  FL  •  then 

CRi  -  J?2)  sin  72  =  J, 
whence 

7?2  _  7?!  -  i—  =  =  &-1  cosec  72.      .     .     (63) 
sin  /2 

Had  2?j  been  required,  the  equation  would  have  been 

J?!  =  #2  -\-l  cosec  72. 
Evidently,  I,  =  I  —  72. 

In  the  field  the  points  77  and  .#  may  be  located  by  running  in 
the  curve  from  A  as  starting-point,  or  run  the  chord 

AF=2Rl  sin  |7 

from  A,  arid  at  T7  deflect  angle1  AFB  —  \I  —  i/2  =  |/i  ,  measure 
FB  —  I  sec  |72  and  BE  —  2,ff2  sin  i/2. 

EXAMPLE.—  A  2°  curve  has  the  P.O.  at  sta.  110,  Ti  =  590  ft., 
T2  =  511.8  ft.,  I  =  30°  50'.     Locate  the  curve. 

By  Table  IX,  T0  =  1580/2  =  790  ft. 

By  formulas  above, 

s  =  200  X  0.85866  =  171.73  ft., 
p  =  200  X  0.51254  =  102.51  ft., 
I  =  790  +  171.73  -  511.8  = 


1QO   51 

tan  7'  =          =  °"  23778  =  tan  25°  40/' 


Then  7,  =  30°  50'  -  25°  40'  =  5°  10'. 

440  97 
7i>2  =  2864.93  -  ~      -'  =  1833  feet. 


82        A    FIELD-MANUAL    FOR    RAIL  HO  AD    ENGINEERS. 

By  Table  1  this  is  seen  to  be  the  radius  of  a  3°  7-J'  curve. 

The  length  of  first  branch  is  258.3  feet,  and  of  the  second  821.3 
feet;  hence  the  P.C.C.  falls  at  112  +  58.3,  while  the  P.T.  is  at 
sta.  120  +  79.6. 

118.  Given  the  Long  Chord  from  P.O.  to  P.T.  of  a  Com- 
pound Curve,  the  Angles  it  makes  with  the  Tangents  and 
One  Radius,  to  Find  the  Other  Radius  and  the  Central  Angles. 

In  Fig.  42  AB  is  known,  as  also  the  angles  HAB  =  a  and 
HBA  =  b.  Two  angles  and  one  side  of  the  triangle  HAB  are 
known,  and  the  sides  HA  =  Ti  and  HB  —  T*  may  be  found, 
after  which  the  solution  is  the  same  as  in  the  last  problem. 

A  solution  may  be  reached  in  a  different  manner.  I  =  a  +  b, 
RAF  =  \I  =  \(a  +  b),  and  BAF  =  $(a  +  b)  -  a  =  \(b  -  a), 
AF  =  2#!  sin  i/.  In  triangle  BAF  two  sides  and  the  included 
angle  are  now  known,  so  AF  and  angle  BFA  may  be  found; 
GFB  =  i/2  =  K  -  BFA> 

Then  EF  =  2Z?j  sin  £/„  , 

and  EB  =  EF  —  BF  becomes  known. 

Then  EB  =  2#2  sin  £/3  =  %&  sin  £/2  -  BF, 

7?  T/* 

Whence  fi,  =  A  _  __     .......    (64) 


Evidently  Ji  =  /  —  J 


119.  Given  the  Radii  and  Central  Angles  of  a  Compound 
Curve  to  Find  the  Tangent  Lengths,  the  Long  Chord  from 
P.C.  to  P.T.,  and  the  Angles  it  makes  with  Tangents. 

In  Fig.  43  draw  AE  and  BE  from  the 
P.C.  and    P.T.    to    the    P.C.C.  ,   then 
calculate  AE  and  BE  by  (16)  or  by 
Table   IX.      In    triangle    AEB   angle 
,          _         AEB  =  180  -  £(/,  +  /„).      Two  sides 
^          ^s      and  the   included   angle  being  known, 
the  triangle  AEB  may  be  solved  for 
AB  and   the  angles  ABE  and  BAE\ 
then 


BAF  = 

FIG.  43.  ABF  —  ABE  +  ^72. 

The  angle  AFB  of  triangle  ABF  now  becomes  known  and,  as 


LOCATION.  83 

AB  is  known,  the  sides  AF  =  2\  and  BF  =  T*  may  be  com- 
puted. 

120.  Given  the  Long  Chord  from  P.C.  to  P.  T.  of  a  Com- 
pound Curve  and  the  Angles  it  makes  with  Tangents  to 
Find  the  Radii  when  the  Common  Tangent  is  Parallel  to  Long 
Chord. 

In  Fig.  43  let  GHl>G  parallel  to  AB,  and  GAB  =  a,  HE  A  =  b 
known.  Then 

BAE  =  EAG  =  GEA  =  \a, 
and  ABE  =  EBH  =  IIEB  =  \b. 

Also,  AEB  =  180°  -  |(a  +  b). 

In  triangle  AEB,  remembering  that 
sin  [180  —  •!(« 


AE= 


. 

sin  i(a  -f-  6) 

and 

^4  #  sin  ^a 


~DTJ1 

JJJL  — 


sin     a 


Since  J.0i  J£  =  a  and  EO*B  =  b,  the  radii  JKi   and  7?2  may  be 
found  from  formula  (16),  or  (16«). 


7?      - 

~  sin  ^6  ~  2  sin  $6  .  sin  £(a  +  6)' 

EXAMPLE.  —  Required  7?i  and  -K2  ,  or  DI  and  J)a  ,  when  AB  = 
900  feet,  a  =  12°,  6  =  15°. 

By  (65),  .K,  =  2407.0  ft. 

By  (66),  R*  =  1543.7  ft. 

From  Table  I,     Z>,  =  2°  22'  50"    and    D9  =  3°  42'  44". 


84        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

B.   Obstacles. 

121.  To  Locate  a  Point  on  uie  Second  Branch  of  a  Com- 
pound Curve  when  the  P.C.G.  is  Inaccessible. 

Ordinarily  the  second  branch  is  located  by  setting  transit  at  the 
P.  G.  C.  and  running  the  curve  from  that  point.  An  obstacle  on 
either  curve  may  then  be  passed  by  the  methods  given  for  simple 
curves. 

When  the  P.C.C.  is  inaccessible, 

,Ap  locate  the  first  branch  from  iheP.C. 

and  the  second  branch  from  the 
P.  T.,  if  this  latter  point  is  known. 
When  this  is  not  the  case  proceed 
by  one  of  the  following  methods: 
FIRST.  By  means  of  a  long 
chord. 

In  Fig.  44  let  E  be  the  P. C. C., 
A  some  known  point  on  first 
branch,  EF  a  tangent  at  E,  and 
AB  parallel  to  FE.  The  station 
numbers  of  A  and  E  being 
known,  the  arc  AE  and  angle  a 


are  readily  found  ;  then 

EL  =  .Ra  vers  b  = 

whence 

vers  a 


vers  a, 


vers  b  = 


(67) 


next, 


AB  —  .#,  sin  ol-f  #2  sin  b  ......    (68) 

Deflect  FAB  -  a  from  tangent  at  A  ;  measure  out  AB  ;  set  the 
transit  at  B  and  locate  the  second  branch. 
BY  TA*BLE  IX.—  Take  the  mid-ordinate  in  table  for  an  inter- 

section-angle 2a  ;  then 


Then  EL  X  D*  is  the  mid-ordinate  for  a  1°  curve  having 
I  —  2b,  from  which  b  becomes  known.  From  the  table  now  find 
AL  and  LB,  the  half-chords  for  angles  2a  and  2b,  and  proceed  as 
before. 

SECOND  METHOD.—  By  means  of  tangents. 
From  Fig.  44,         AF  —  FE  =  Jti  tan  \a. 


LOCATION. 


85 


Set  transit  at  F,  deflect  GFE  =  a,  and  by  some  indirect  method 
measure  to  an  accessible  point  H. 

EH=  FH-  FE, 
and 


tan 


=  ——,  from  formula  (14). 


Angle  b  is  now  known  and  equals  GHE,  which  deflect  from 
EH;  then  measure  HB  =  EH,  and  with  transit  at  B  locate  the 
second  branch  of  curve. 

OR  BY  TABLE  IX.  —  Find  AF  =  FE,  the  tangent  distance  for 
I  —  a;-  then  having  EH  measured,  take  Ti  =  EH  X  Di  and  find 
the  corresponding  angle,  which  equals  b  ;  then  proceed  to  locate 
curve  as  above. 

EXAMPLE.—  Let  A  be  at  sta.  126,  P.  C.  C.  at  128  -f  25;  the  degree 
of  first  branch  4°,  and  of  second  6°. 

By  the  first  method  EL  =  17.635  for  a  =  9°,  and  b  =  11°  2', 
nearly.  AL  =  224.1  ft.,  BL  —  182.75  ft.,  and  therefore  AB  = 
406.85  ft.  Angle  b  =  11°  2'  corresponds  to  183.9  ft.  around  6° 
curve;  hence  the  P.T.  number  is  130  -+-  08.9. 

By  the  second  method  AF  =  112.74  ft.  Suppose  FH  =  264  ft., 
then  EH=  151.26  ft.,  which  multiplied  by  6  gives  907.56  ft., 
corresponding  to  /  =  18°.  The  arc  EB  is  now  300  ft.,  making 
B  fall  at  sta.  131  +  25. 


C.     Change  of  Location. 

122.  Having  a  Simple  Curve  Located  to  Find  the  P.  C.  C.  so 
that  a  Curve  of  Given  Radius  shall  connect  with  a  Given 
Tangent    Parallel     to    Tangent     to 
Located  Curve. 

Let  NAB,  Fig.  45,  be  the  located 
curve,  HF  the  tangent  in  which  the 
second  branch  must  end.  The  dis- 
tance BG  =  p  between  tangents  is 
known  from  measurement.  If  angle 
a  can  be  found,  the  arc  BA  becomes 
known  and  the  point  A  can  be  located 
from  B.  Draw  0*L  from  the  center 
of  second  branch  perpendicular  to 
0|jB>  Iu  trjang]e  Q^L,  0,0,= 
-f  p)  ;  therefore 


Rl  - 


-  R1  - 


86        A    FIELD-MANUAL   FOR   HAILROAD    ENGINEERS. 


cos  a  = 


R\  — 


Then  a  divided  by  Di  gives  arc  1L4. 

If  desired,  BH  may  be  found  from  the  right  triangle  BHO,  in 
which  the  side  BG  =  p  and  angle  OHB  =  ^a  are  known— - 
A,  H,  and  B  lying  in  the  same  straight  line  ;  then 


BH  =    .    ,     =  p  cosec  la. 
sin  Aa 


(70) 


Or  BA  and  #J.  may  be  found  from  Table  IX,  after  which 
BH=BA-  HA. 

EXAMPLE. — A  3°  curve  ends  in  a  tangent  at  sta.  160  -f-  50, 
35  ft.  outside  of  desired  tangent.  Find  the  point  of  compound- 
ing with  a  4°  50'  curve. 

From  Table  I,  R  for  3°  curve  equals  1910.08  ft.,  and  for 
4°  50'  curve  1185.78  ft. 


Then,  by  (69),   cos  a  =  1  - 


35 

7243 


=  0.95168. 


From  table  of  cosines  angle  a  is  found  to  be  17°  53'.  Dividing 
this  by  3  gives  5.961  stations  for  the  arc  BA.  Hence  the  P.C.C. 
number  is  160.50  -  5.961  =  sta.  1 54  -f  53.9,  and  the  new  P.T.  is 
at  sta.  158  +  23.9. 

123.    Given    a    Located    Compound    Curve    ending    in  a 
Tangent  Parallel  to,  and  a  Given  Distance  from,  a  Tangent 
in  which  the  Curve  is  required  to  end.     To  Find  the  Neces- 
sary Change  in  P.  C.  C. 
FIRST  CASE. — Terminal  branch  having  shorter  radius. 

In  Fig.  46  let  ABC  be  the  located 
curve,  AEF  the  one  required  ;  angle 
BOiC  =  a  known,  and  also  MN  =  p. 

If  angle  EOM =  b  can  be  found,  the 
angle  of  retreat  from  B  to  E  will  equal 
b  -  a. 

Draw  O/7f  and  OiL  perpendicular 
to  ON,  which  is  parallel  to  0,C. 


N 


FIG.  46. 


Then     OK  =  (R  —  R,}  cos  b, 
OL  —  (R  —  lit)  cos  a. 


LOCATION. 

Now  LM  =  Rt  -  KL  -  R,  -  MN,  from  which  KL  =  MN  =  p. 
Hence 

(E  —  Ri)  cos  b  =  (R  —  Ri)  cos  a  —  p. 
From  which 

cos  b  =  cos  a  —  —  -  —  ......     (71) 

It  —  H\ 

Divide  b  —  a  by  D,  the  curvature  of  first  branch,  and  move 
back  that  number  of  stations  from  B  to  the  new  P.  C.C.  at  E. 

Join  0,0,';  evidently  FC  =  O.O/,  and  angle  JiO.'O,  =  CFG  ; 
00,'  0,  =  90°  -  $(b  -  a),  00,'  K  =  90°  -  b.  Hence 

CFG  =  #0/0!  =  [90  -  i(b  -  a)]  -  (90  -  6)  =  i(b  +  a).     (72) 
From  triangle  CGF, 

8  +  a>-    '   '   (73) 


Or,  from  triangle  00/0,  , 

^(7  =  0/0,  =  2(R  -  R,)  sin  ±(b  -  a). 


Had  J.^F  been  the  original  curve,  b  would  have  been  known 
and  a  required. 

From  (71),  cos  a  =  cos  b  -f  _    P  _  .  .  (74) 

.a  —  it, 


angle  CFIfnte  given  by  formulas  (73)  and  (72). 
EXAMPLE.— A  2°  curve  compounds  with  a  4°  curve  at  sta. 
82  -f  30;  a  =  20°  30',  p  =  40  feet.     Find  number  of  new  P.C.C. 
and  distance  between  P.2\s. 

40 
From  (71),    cos  b  =  0.93667  -  2864  9  _  1433  7  =  0'90874- 

This  yields  b  =  24°  20',  and  b  -  a  =  3°  50'. 
The  change    in  P.C.C.  is  ^-8  =  1-917  stations;    the  P.C.C. 
number  is  therefore  82.30  -  1.917  =  sta.  80  -f  38.3. 


88        A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 


By  (72),  CFG  =  |(24°  20'  +  20°  30')  =  22°  25'. 

By  (73),  FC  =  40  X  2.62234  =  104.9  feet. 

SECOND  CASE. — The  terminal  branch  having  longer  radius. 
Let  CAB,  Fig.  47,  be  the  located 
curve  with  P.C.C.  at  At  and  let 
FK  be  the  tangent  in  which  the 
curve  is  required  to  end. 

The  distance  BK  =  p,  the  radii 
OA  =  R,  0,A  =  R, ,  and  angle 
AO,B  =  a  being  known,  it  will 
be  sufficient  to  find  angle  EO,'F 
in  order  to  get  the  angle  of  ad- 
vance, AOE  —  a-  b.  Draw  OL 


Fra.  47. 
Oi'OM and  0>OL, 


and   OiN  perpendicular    to  -Oi'F 
and    0,B.      From    the    triangles 


(R,  -  R)  cos  b  =  0/.ZV-J-  (Rl  -  R)  cos  a. 
But  Oi'N  =  KB  =  p ;  therefore 

(R!  —  R)  cos  b  =  p  -\-  (R,  —  R}  cos  a. 
P 


Whence 


cos  b  =  cos  a  -f- 


(75) 


Then  — =—  will  be  length  of  curve  from  A  to  E. 


Angle  KFB  =  N0,0,'  =  00,0,'  -  N0,0. 
But   00, 0,'  -  90°  -  l(a  -  b)    and    N0,0  =  90  -  a. 

'.    KFB  =  [90°  -  l(a  -  b)]  -  [90  -  a]  =  K«  +  ft)- 
From  triangle  KFB, 

FB  =  siu  i(a  i   ft)  =  p  •  cosec  *(a  +  &)'   '     '    •     (76) 

Or,  from  triangle  0,00,',  since  0,0,'  =  FB, 
FB  =  2(R,  -  72)  sin  |(o  -  b). 


LOCATION. 


89 


If  AEF\\n<\  been  the  located  curve,  b  would  have  been  given 
and  a  required.     From  formula  (75), 


cos  a  =  cos  6  —  TT- 


P 


It,  -  R' 


(77) 


EXAMPLE. — A  5°  curve  compounds  at  sta.  60  with  a  2°  curve, 
and  the  P.T.  is  at  sta.  80.  What  will  be  the  number  of  P.C.C. 
if  the  P.T.  fall  in  a  tangent  81  feet  inside  of  terminal  tangent? 
Here  a  =  40°. 


81 


By  (75),        cos  b  =  0.76604  +  -^  =  0.81316. 


1719 


Hence  b  =  35°  36'  and  a  —  b  =  4°  24',  corresponding  to  220 
feet  around  the  2°  curve.  The  number  of  the  new  P.C.C.  is 
therefore  62  +  20, 

angle  KFB  =  £(40°  0'  +  35°  36')  =  37°  48', 


and 


FB  =  Sl  X  1.63157  =  132.16  feet. 


124.  Given  a  Located  Compound  Curve  to  Find  Necessary 
Change  in  P.  C.  C.  and  Radius  of  Second  Branch  to  make  the 
P.T.  fall  in  a  Tangent  Parallel  to  First  Terminal  Tangent 
and  in  a  Point  on  the  Same  Radial  Line. 

FIRST  CASE.— Second  branch  having  shorter  radius. 

In  Fig.  48,  OB=R,  01B=R,  angle 
a  and  HO  =  p  are  known.  O^E=R^ 
and  angle  b  must  be  found  ;  then 

— —  =  BE  will  be  the  change  in 

P.C.C. 

Produce  first  branch  to  K,  where 
OK  is  parallel  to  0,  C.  Since  BOK 
=  BO,  C,  B,  K,  and  C  lie  in  the  same 
straight  line;  and  since  EO^F  — 
EOK,  E,  F,  and  K  lie  in  the  same 
straight  line.  Therefore 

{a,     and    KFH=$b. 


FIG.  48. 


90        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

From  triangles  KFH and  KCG, 

HK       OK       OH  p 


But  FH=  0i  Z  =  (B-Bi)  sin  a. 


.'.     tan  ^b=  tan  \a 


P 


(R-  A)  sin  a' 
From  triangles  00,  L  and  002 M, 

(R  —  Rz)  sin  b  =  (R  -  Ri)  sin  a. 


When 


(78) 


(79) 


Had  ^l^F  been   the  first  curve  located,  5  and  ^?2  would  be 
known,  a  aud  J?i  required. 
From  the  figure,  reasoning  as  before, 


tan  \a  — 


P 


i (80) 


and 


SECOND  CASE.—  Second  branch  having  longer  radius. 


FIG 


(81) 


In  Fig.  49  let  AB  be  the  located  curve,  JS'jP'the  curve  required, 
OA  =  R,  0,A  -  Rlt  0*E  =  A,  FB  =  p. 


LOCATION.  91 

7?a  and  angle  b  are  wanted,  angle  a  being  known. 
We  can  show,  as  in  first  case,  that 

HFK  =  lb,     HBL  =  \a, 

OM  =  KF  =  LB  =  (/?,  -  E)  sin  a; 
and  hence 

HL     HK  .    p 
t™y>=Bl=j£+2L. 

Or  inserting  values, 

tan  Ib  =  tan  \a  -f  — ——} — (82) 

(#1  —  E]  sin  a 

Angle  b  now  becomes  known  and  — ~ —  =  -A-Z?  in  chains,  which 

is  the  change  in  position  of  P.C.C. 
From  triangles  OOiJl/aud  002Jf, 

(JS2  -  R)  sin  6  =  (^  -  7?)  sin  a 


Had  the  new  tangent  fallen  outside  the  old  one,  we  should  have 
had 

.     .     .     (84) 


VC*M     Tk  \M     — —      Ul*«-l     -K1S  /    •»•»  T»  •  »» 

(#2  —  .R)  sin  6 
and 


(85) 


125.  Having  a  Located  Compound  Curve,  to  Find  the 
Change  in  P.C.C.  and  Radius  of  Second  Branch  in  order  to 
Cause  P.  T.  to  Fall  at  a  New  Point  in  Terminal  Tangent. 

FIKST  CASE. — Second  branch  having  shorter  radius. 


92        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

In  Fig.  50  let  NAB  be  the  located  curve,  and  C  the  point  where 
P.T.  is  required  to  fall.  Let  BG  =  k,  OA  =  R,  0,B  -  JR,,  and 
angle  0\ OH  =  a  be  known;  angle  b  and  R?  are  required. 


Extend  first  branch  to  F,  making  OF  parallel  to  OiB.  A,  B, 
and  Flie  on  a  straight  line,  for  angles  AOiB and  AOF&re  equal; 
likewise  E,  C,  and  F  lie  on  the  same  straight  line. 

From  triangles  GBFznd  OOF, 

OB      CB  k 


But      OF  -  HM  =  (R- 


. :  cot  46  =  cot  \a  — 


cos  a)  =  (R  —  Rt)  vers  a. 
k 


(R— Hi)  vers  a 
From  triangles  OOiH  and  00*L,  since  0,P  =  k, 
(R  -  /?,)  sin  b  =  (R-  -K,)  sin  a  -  A:. 

Whence 

A;  -  (R  -  Ri)  sin  o 


(86) 


(87) 


Then  b  —  a  divided  by  D  gives  arc  A  K    With  radius  R?  locate 
the  curve  EC  from  C  or  JK 


LOCATION. 


93 


Had  NEC  been  the  located  curve,  R,  _Z?2 ,  and  6  would  have 
been  known,  Ri  and  a  required.     In  this  case 


cot  \a  =  cot  \b  — 


,      ....     (88) 


(R  —  Rt)  sin  & 


(89) 


SECOND  CASE. — Terminal  branch  liamng  longer  radius 

In  Fig.  51  let  NAB  be  the  located  and  NEC  ike  required  curve. 


Fia,  51. 


Let  CB  —  k  be  known.     Then,  as  in  the  first  case, 


GC  _QB        k 
-~-_  —  _— . 


Ri  -  R)  versa 
and  (Rt  -  R)  sin  a  =  (R*  -  R)  sin  b  -f  k  ; 


whence 


(Rl  —  R)  sin  a  —  k 
sind        ~~* 


.      90) 


(91) 


94        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

Had  NECbuen  located  and  NAB  required,  the  equations  would 
have  been 


and 


cot  \a  =  cot  |6  + 

in. 
&  = 


#a  -  R)  vers  b  ' 
-  R)  sin  b  +  k 


sin  a 


(92) 
(93) 


In  either  of  these  two  cases  if  k  is  unknown  and  the  new  radius 
given  or  assumed,  the  desired  angle  and  the  value  of  k  may  be 
found  from  the  foregoing  equations.  Or,  knowing  the  new  angle, 
the  new  radius  and  value  of  k  may  be  found  from  the  same 
equations. 

126.  To  Replace  a  Curve  of  Given  Radius,  which  unites 
Two  Tangents  with  Known  Intersection-angle,  by  a  Three- 
centered  Compound  Curve. 

In  Fig.  52    let  OA  =  R   be   the   radius  of    located  curve, 


02  (7=  Oi'A  =  Rv  the  radius  of  terminal  portions  of  the  three- 
centered  curve,  and  the  other  notation  as  shown  in  the  figure. 

Draw  0202',  and  draw  FOH perpendicular  thereto.     From  tri- 
angles O^H  and  0*OH, 

0*H  =  (R,  -  R,)  sin  i/t  =  (R*  -  R)  sin  |I.  .    .    .     (a) 

Suppose  It*  and  Ri  to  be  assumed  ;  then  equation  (a)  yields 


'     (94) 


LOCATION.  95 

Then  AO*E  =  CO*G  =  \(I  -  /,).      .     .     .     (95) 

Suppose  AOi'E,  CO^G,  and  7?2  to  have  been  assumed.     From 
(95)  find  /i  ;  then,  from  equation  (a), 


(96) 


EXAMPLE.  —  Given  a  4°  curve,  /  =  38°,  and  the  terminal 
brunches  composed  of  a  2°  curve  for  two  stations,  to  find  Ri  and 
Di  for  the  central  portion. 

Here  L  =  38°  -  2(2  X  2)°  =  30°. 

From  Table  I,  R*  =  2865  ft.,     R  =  1432.7  ft. 
Whence  Ri-R=  1432.3  ft. 

Log  1432.3       =  3.15603 
"    sin  19°  0'  =  9.51264 


2.66867 
sin  15°  (X  =  9.41300 


.-.     log  1801. 1  =  3.25567 

Therefore  R,  =  2865  -  1801.7  =  1063.3  ft.,  and,  by  Table  I, 
Dl  =  5°  23'. 4,  nearly  enough. 

127.  To  Substitute  a  Curve  of  Given  Radius  for  a  Tangent 
uniting  Two  Curves. 

In  Fig.  53  let  the  tangent  BC  =  t,  OB  =  R,  01C=R1,  and 
0-iA  —  Ri  be  known. 

Angles  a,  I,  and  c  must  be  found  in  order  to  substitute  curve 
AE  for  the  system  ABCE. 

Draw  OF  parallel  to  BC,  then  O^F=  Rt  —  R,  and,  from  triangle 
OOtF, 


00,  =  -.-   -, ;  =  t.  cosec  d  =  \/(R,  -  R)*  -f  <».      .     (98) 
sin  fl{ 


96        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

Now  in  triangle  00\0<i  three  sides  are  known  aiid  the  angles 
c  and  e  may  be  computed.    Thus  if  s  is  the  half-sum  of  the  sides, 


Angle  e  may  be  found  in  like  manner,  then  b  =  180°  —  (e  ~\-  d), 
and  a  =  c  —  b. 

Points  A  and  JSTmay  now  be  located  and  the  curve  traced. 

EXAMPLE. — A  3°  and  a  5°  curve  are  united  by  a  tangent  500 
feet  long.  Replace  by  a  2°  curve. 

Here         K,  -  R  -  1910  -  1146  =  764  feet. 

500 
By  (97),        tan  d  =  —  =  0.65444  =  tan  33°  12' 

By  (98),          00,  =  913.1  feet. 

In  triangle  OOiO^,  00l  =  913.1,  OiOa  =  954.9,  and  00a  = 
1718.7  feet.  Solving  for  e  and  c, 

e  =  133°  36',     c  =  23°  0'.     Then     &  =  13°  12',     a  =  »°  56'. 
ARTICLE  10.    TRACK  PROBLEMS. 

128.  Reversed  Curves  should  never  be  employed  on  main 
lines  because  of  the  shock  due  to  sudden  reversal  of  curvature 
and  superelevation  of  outside  rail.  A  short  tangent  should  be 
interposed  between  the  two  curves,  which  may  ordinarily  be 
done  by  changing  the  end-points  of  the  curve,  or  slightly  altering 
the  radius.  If,  however,  transition  curves  are  employed  to  ease 


LOCATION. 


97 


off  both  curves,  there  would  seem  to  be  no  objection  to  the  use  of 
curves  of  contrary  flexure,  provided  the  track  may  be  kept 
always  iii  perfect  condition.  In  yards,  crossovers,  and  where 
connection  is  made  with  existing  track,  reversed  curves  may  be 
employed,  and  are  often  imperative. 

129.  Having  a  Located  Curve  Intersected  by  a  Straight 
Line,  to  Connect  them  by  Another  Curve. 

Either  the  radius  of  the  joining  curve  may  be  given,  or  else  the 
point  on  first  curve  at  which  the  junction  must  be  made.  The 
angle  between  a  tangent  to  located  curve  at  the  point  of  meeting 
and  the  straight  line  must  be  measured.  Four  possible  cases 
occur. 

FIRST  CASE. — Joining  curve  tangent  to  located  curve  internally 
and  on  same  side  of  cutting  line  as  center. 

In  Fig.  54  let  GF  be  joining  curve,  with  center  Oi  and  radius 
EL  Let  radius  of  located  curve  OF  =  E.  Draw  O^G  and  OH 
perpendicular  to  the  cutting  line  produced,  and  O^K  parallel  to 
AH.  If  Ri  is  known,  we  must  determine  angle  b,  a  having  been 


FIG.  54. 


measured;  then  b  —  a  gives  the  length  of  arc  from  A  to  F where 
the  P.C.C.  is  to  be  located.     In  the  triangle  KOOi  we  have 

OK  =  OH  -  R,    and     00,  =  R  -  R,. 


Then 


cos  b  = 
b-  a 


R  cos  a  —  Rl 


R-Rl 

=  arc  AF. 


(99) 


Had  Fbeen  given,  we  should  have  b  —  a -\-AOF,  and,  from  (99), 
R  (cos  a  —  cos  b}      R  (cos  a  —  cos  b)       . 

Si  i  J==    — =: __„„ .       (1UU) 

1  ,—  cos  b  vers  b 


98        A    FIELD-MANUAL   FOE   RAILROAD    ENGINEERS. 


EXAMPLE. — A  1°  curve  is  cut  by  atangcut  that  makes  au  angle 
of  64°  32'  with  tangent  to  curve.  Unite  by  means  of  a  4°  curve. 

By  (99),  cos  b  =  0.24000  =  cos  76°  07',  and  therefore  b  -  a  = 
11°  35',  making  AF,  of  figure,  11.58  stations. 

SECOND  CASE.  —  Joining  curve  tangent  internally  to  located 
curve  but  on  opposite  side  of  cutting  line  from  center  of  located 
curve. 

In  Fig.  54  let  arc  ME,  with  center  02  and  radius  1?2 ,  be  the 
joining  curve.  From  the  figure, 


cos  d  = 


H  cos  a  -\-  J?2 
R-R* 


(101) 


Then  arc  AE  =  a  -  d  divided  by  D,  and  c  =  180°  -  d. 
Had  the  point  E  been  given   and  JR3  required,  it  would  have 
been,  from  (101) 

7?(cos  d  —  cos  a) 


It*  = 


(102) 


1  -j-  cos  d 

EXAMPLE. — Take  the  same  example  as  in  first  case.     Here, 
By  (101),        cos  d  =  0.9068  =  cos  24°  56'. 
Then  64°  32'  -  24°  56'  =  39°  36', 

equivalent  to  39.600  stations  around  curve  from  A  to  E. 

THIRD  CASE. — Joining  curve  tangent  externally  to  located  curve, 
with  center  on  same  side  of  cutting  line. 

Lr 


FIG.  55. 


In  Fig.  55  let  arc  5(7,with  center  0,  and  radius  Ri ,  be  the  join- 
ing curve.  Draw  0,E  parallel  to  CF,  aud  OjCand  OF  perpen- 
dicular thereto. 


LOCATION.  99 


From  the  figure, 

(R  +  RJ  cos  b  =  R  cos  a  - 
R  cos  a  — 


(103) 


Then  d  =  180  —  b,  and  AOB  =  b  —  a.     The  curve  may  now  be 
traced  on  the  ground. 

If  AC  is  wanted,  we  have  AC  =  (R  +  R\)  sin  b  —  R  sin  a. 

If  the  point  B  is  fixed  and  R!  required,  there  results,  from  (103), 

=  B  (cos  »-  cos  » 
1  +  cos  6 

EXAMPLE.  —  Take  the  example  given  for  the  first  and  second 
cases 
By  (103), 

5730x0.43-1432.5 
COS  b  ^   —5780  +  1488.6         =  °'144  =  «»  81    ^ 

b  -  a  =  81°  44'  -  64°  32'  =  17°  12',  equivalent  to  17.2 
stations  on  located  curve  from  A  to  B.   Angle  d  =  180°  —  81°  44' 
=  98°  16',    equivalent   to  24.567  stations  from  B  to  C  on  the 
4°  curve. 

FOURTH  CASE.  —  Joining  curve  tangent  externally  to  located  curve, 
with  center  on  opposite  side  of  cutting  line. 

Let  0a,  Fig    55,  be  center  of  joining  curve,  R*  its   radius. 
From  the  figure, 

(R  +  /?„)  cos  c  =  R  cos  a  +  R2. 
R  cos  a  +  7?2 

•••cosc=     U  +  K     ......   •   •  •  •   (105> 

If  Mis  fixed  and  R2  required,  (105)  yields 

„        .K(cos  c  —  cos  a)       7?(cos  c  —  cos  a) 

J-12  =  -  ;  -  .  .       (1UO) 

1  —  cos  c  versm  c 

EXAMPLE.  —  Take  same  example  as  in  preceding  cases. 
By  (105),  cos  c  =  0.54403  =  cos  57°  02'. 

Then  a  -  c  =  64°  32'  -  57°  2'  =  7°  30', 


100     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


calling   for   a  distance   of   7.50  stations  from  A  to  M  around 
1°  curve.     From  Mto  H  011  4°  curve  is  14.258  stations. 

130.   To  Locate  a  Y 

A  Y  is  made  up  of  a  system  of  tracks  so  arranged  as  to  admit 
of  turning  an  entire  train.  Three  of  the  most  used  arrangements 
are  given  below. 

FIRST  CASE. — One  branch  of  Y  a  straight  line. 

This  is  only  the  special  case  of  the  last  problem  in  which  the 
cutting  line  becomes  tangent  to  both  curves.  In  Fig.  56,  if  any 


FIG.  56. 


one  of  the  points  A,  B,  or  G  is  given,  the  others  may  be  located 
by  finding  the  angles  c  and  6.  Draw  O^E  parallel  to  CA  ;  then 
in  triangle  OOiE 

RJcosb  =  R  -  EL 


.  •.  cos  b  = 


R- 


(107) 


This  follows  at  once  from  (103)  by  making  angle  a  —  0.  Then 
angle  c  =  180  —  b.  If  AB  were  a  located  curve  and  the  point 
B  given,  formula  (107)  would  furnish  us  a  value  for  Rt. 

Another  solution  is  to  produce  the  tangent  at  B  to  cut  AC  at  F; 
then  AF  =  FG  =  BF.  Join  F  with  0  and  0,  ;  it  can  easily  be 
seen  that  angle  OFOi  =  90°,  and,  by  geometry, 


BF= 


Therefore 


BF 

— 


R, 


(108) 
(109) 


and 


=  "Jr~4/£ (HO) 


LOCATION. 


101 


EXAMPLE.— Let  AB  be  a  3°  curve,  BC  a  6°  curve,  the  point  A 
at  station  180. 
Tty  (107), 


The  number  of  B  is  180  +  23.511  =  203  +  51.1.     Angle  e  = 
109°  28',  equivalent  to  18.244  stations  on  the  6°  curve. 

SECOND  CASE.  —  The  three  Ranches  curved  and  convex  towards 
each  other. 

Given^the  three  radii  and  any 
one  of  the  points  A,  B,  or  0, 
Fig.  57,  we  have  only  to  find  the 
angles  at  the  center,  then  divide 
these  angles  by  the  degrees  of 
the  respective  curves  to  get  their 
lengths  and  locate  the  three 
branches. 

In  the  triangle  00i02,  letting 
00,  =  I,  0,0a  =  m,  002  =  n, 


FIG.  57. 


we  shall  have,  by  trigonometry, 


cos  ia  = 


+  R,  + 


(111) 


Angles  b  and  c  may  be  found  in  like  manner. 

The  angles  may  be  found  otherwise  by  letting  fall  a  perpen- 
dicular from  one  vertex  upon  the  opposite  side,  as  OE  perpen- 
dicular to  0,  02.  Then  from  the  relation 

0i02  :  0,0  +  002  =  OOi  -  002  :  OtE  -  0*E 

determine  02#  and  O^E\  then  the  right  triangles  O^OE  and 
OiOE  yield  values  of  cosine  a  and  cosine  c,  after  which  b  may 
readily  be  obtained. 

THIRD  CASE. — One  brancJi  concave  to  the  other  two. 

In  Fig.  58  the  triangle  00, 02  may  be  solved  for  the  angles  at 
0,  0i,  and  02 ;  for  if  the  radii  are  given,  the  sides  00,  =  R  -  Hi, 
002  =  R  —  Rz,  and  OiOj  =  Rt  -\-  R2  are  known  and  the  solution 


102     A   FIELD-MANUAL   FOR  RAILROAD   ENGINEERS. 

is  the  same  as  for  second  case.  Then  b  is  the  central  angle  for 
curve  AB,  a'  —  180  —  a,  the  central  angle  for  AC,  and  c'  = 
180  —  ct  the  central  angle  for  curve  BC. 


EXAMPLE.—  If  A  is  at  sta.  820  on  the  1°  curve  AB,  AC  an 
8°  curve,  connect  with  a  6°  curve  CB.     Here  we  have 

0*0  =  5730  -  717  =  5013,    OiO  =  5730  -  955  =  4775, 
and  0,0,  =  955  +  717  =  1672. 

Solving   this  triangle,   we    get    c  =  88°  20',    b  =  19°  28',    and 
a  =  72°  12'.      The  number    of  B  is   therefore  820  +  19.467  = 

91  667 
839  -f  46.7  ;  the  length  of  CB  is  —  ^—  =  15.278   stations,    and 

1  07  8 

of  A  C  is  —    —  =  13.475  stations. 
8 

131.    To    Locate    a    Reversed     Curve    between    Parallel 
Tangents. 

FIRST  CASE.  —  Radii  equal. 

(a)  The  equal  radii  R  and  distance  p  between  tangents  known. 
In  Fig.  59  draw  OE  parallel  to  AG  to  meet  OiB  produced. 
From  triangle  OEOi, 


and  OE  =  2R  sin  a  .........     (113) 


LOCATION. 


From  triangle  ABG, 


AB  =  — — — -  =  p  cosec  \a  = 


103 


FIG.  59, 

(b)  AG  and  p  known,  M  required. 

Here  AB  =  \/AG-  +  p2  =  k.  Draw  OJ2"  to  the  mid-point  of 
AC.  Triangles  AOH  and  ABG  are  similar  and  AH '=  \k, 
Therefore 

?L  -A. 

i*  ~2>' 


whence 


(115) 


EXAMPLE. — Connect  two  parallel  tracks,  30  ft.  c.  to  c.  by  a  7° 
reversed  curve.     From  Table  I,  R  =  819  feet,  and,  by  (112), 


cos  a  =  1  — 


30 
1638 


=  0.98167  =  cos  10°  59'. 


By  (113),     OE  =  1638  X  .19052  =  312.1  feet. 


By  (114),     AB  —  |/(312.1)2  +  (30)2  =  313.5  feet. 

If  p  =  30,  OE  =  312.1,    or  AB  =  313.5  had  been  given,  we 
should  have  had,  by  (115) 


104     A    FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

SECOND  CASE.  —  Radii  unequal. 

(a)  Suppose  the  radii  R  =  0  A  and  R,  —  O^B  (Fig.  59)  to  be 
known  We  must  find  central  angle  a  and  AB  —  k.  From  the 
triangle  OO^E, 


Then  AB  will  be  given  by  (114). 

(b)  Suppose  AB  =  k,  p  and  R  known,  to  find  Ri  and  angle  a. 

Triangle  ABG  yields 

sin  \n  =  -f-  .........     (117) 

K 

OiLB  is  similar  to  AGE.    Hence 

1*L-  A 

LB~  p' 

But  AC  =  2#  sin  \a,  and  LE  =  i(k  -  AC)  -  $Ci.     Inserting 
this  value  of  LB  and  solving  for  Rit 


«=1 

From  similar  triangles, 


R  ~  k  -  0,  ' 

Inserting  me  value  of  Ci  =  ^-~  -  from  (118)  and  solving  for 
j  ,  we  get 


EXAMPLE.—  AB  =  300',    p  =  30',    #  =  819  ft.,  to  find  angle 
a  stud  JBi. 

30 

By  (117),         sin  \a  =  —  ^  =  0.10000  =  sin  5°  44'. 

Therefore  angle  a  =  11°  28'. 
By  (119),  Ri  =  -8         -  819  =  681  ft.  ,  an  8°  25'  curve. 


LOCATION. 


105 


132.  To  Connect  Two  Parallel  Tracks  by  a  Crossover  com- 
posed of  two  Z)°  Curves  with  a  Given  Length  of  Tangent 
between  Points  of  Contrary  Flexure. 

In  Fig.  60  let  AFOB  be  the  re- 
quired crossover,  FG  =  l,  EB=p, 
and  OA  =  OB  —  R  known; 
angle  a  and  AE  =  x  are  re- 
quired. 

Draw   OM  parallel  to  AE  to 
meet  O'B  produced ;    draw  also      R 
OC  parallel    and  equal  to  FG ; 
join    0  and   0'.     From  triangle 
OO'C, 

tany=;rL  (120) 


&£*t3 


FIG. 


00'  =  -   --  = 
cosy 

Then  in  triangle  OO'M, 

O'M 

COBZ  =  W  = 

Now  knowing  y  and  z, 


(121) 

(122) 

(123) 
(124) 


Next,  x  =  OM=  00'  sin  z  =  2R  sec  y  sin  z. 

EXAMPLE.— Given  D  —  7°  30',  p  =  62  ft.,  I  =  100  ft.,  to  locate 
crossover  when  A  is  at  sta.  86  -f-  20. 

By  (120), 

log  tan  y  =  2  -  3.18441  =  8.81559  =  log  tan  3°  44'. 
By  (121), 

log  00'  =  3.18441  -  9.99908  =  3.18533  =  log  1532. 
By  (122), 

log  cos  z  =  3.16643  -  3.18533  =  9.98110  =  log  cos  16°  47'. 
By  (123), 

a  =  16°  47'  -  3°  44'  =  13°  3'. 
By  (124), 

log  x  =  3.18533  +  9.46053  =  2.64586  =  log  442.4. 


106     A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 


133.  To  Find  the  Radius  of  the  Reversed  Curve  AFE,  Fig. 
61,  Given  Angles  /  and  1' ',  and 
BG  =  k. 

From  the  figure, 


FIG.  61. 


T=  CF. 
Adding, 
JF?(tan  \1  -f  tan  i/')  =  BC  =  k. 


Whence 


R  = -=—       — - (125) 

tan  £/  +  tan  \I' 

EXAMPLE.— Given  /  =  10°,  1'=  20°,  BC  =  700  feet,  to  find  R. 
700 


By  (125),     R  = 


0.08749  +  0.17633 


35-  =  2653  ft.,  a  2°  9V  curve. 


134.  To  Locate  a  Reversed  Curve  between  Fixed  Points. 
In  Fig.  62  let  AB  =  k,  and  angles  1  and  /'  be  known.      We 


FIG.  62. 

have  to  find  R  and  the  angles  a  and  b. 

Draw  O'O  parallel  to,  and  OG  and  O'F  perpendicular  to,  AB. 
Angle  AOG  =  I  and  BO'F  =  /'.  Then  OE  =  R  cos  /  and  O'F 
=  R  cos  /'.  Hence 

O  =  7?(cos  I  +  cos  /')• 
In  triangle  00' O,  00'  =  2#.     Therefore 

R(cos  1  -r  cos  1 ')      cos  7  +  cos  / ' 


.     (126) 


an  expression  from  which  R  has  disappeared. 


LOCATION". 


107 


We  now  have  a  =  I  -|-  x    and     b  =  I'  -{-  x. 

To  find  Ewe  have  AE  +  EF  +  FB  =  k, 
or  R  sin  /  -f-  2R  sin  x  +  2?  sin  /'  = 

Whence 


(127) 


sin /  + sin  1'  +  2  sin  x" 

Another  expression  for  R  can  be  found  by  drawing  AN  and  BL 
perpendicular  to  00',  and  BN  parallel  thereto.  Then,  since 
4  BAN  =  x, 

R  sin  a  -f-  R  sin  b  =  k  cos  x. 


(128) 


..         . 

sin  a  -)-  sm  b 
EXAMPLE.  —  Take  the  example  of  the  last  problem, 

k  =  700,     7=10°,     J'  =  20°. 
By  (126), 

cos  x  =  1(0.98481  -f  0.93969)  =  0.96225  =  cos  15°  48'. 
We  now  have        a  =  25°  48'     and    b  —  35°  48'. 

B  -  =  660-2  ft-  •  -  8° 

135.  To  Connect  Two  Divergent  Tangents  by  a  Reversed 
Curve. 

FIRST  CASE.  —  Advancing  towards  the  P.I. 

Given  the  radii  R  and  Rl  ,  the  angle  7  and  AC  =  &,  to  find  the 
angles  a  and  6  (Fig.  63). 


o, 


JQ 


FIG.  63. 

Draw  OO  parallel  to  the  tangent  BO  to  meet  O^B  produced. 
Then  EF  =  BO  =  AF-  AE, 

Therefore  BG  -  R  cos  /  -  k  sin  /. 


108     A   FIELB-MANUAL   FOR   RAILROAD    ENGINEERS. 
From  trianle  OOiG 


Then  a  =  MO*N  =  b  -  I,  0>M  being  parallel  to  OA. 
SECOND  CASE.—  Receding  from  the  P.  I. 

In  Fig.  63  we  have  EC  =  ki  ,  angle  I,  R,  and  Ri  given,  to  find 
angles  a  and  b. 
Produce  OA  to  meet  OiL  drawn  parallel  to  GA.     AL  equals 


0,H  =  Rl  -  HB=  R,  -  k,  tan  1. 

.'.    AL  =  0^  =  (Rl  -  fa  tan  /)  cos  /. 
Hence 

OL  =  R  +  (#,  -  fa  tan  7)  cos  /  =  R  +  Rl  cos  I  -  fa  sin  /. 
From  triangle  OOiL, 


cos  a  = 


OL 
00, 


Evidently, 


R  +  R,  cos  7  -  fa  sin  / 


b  =  a       /. 


(130) 


136.  To  Change  the  P.R.C.  so  that  Second  Branch  of 
Curve  shall  End  in  a  Tangent  Parallel  to  Terminal  Tangent 
and  Distant  p  therefrom. 

In  Fig.  64  let  MAB  be  the  located  curve,  EN  =  p.    We  must 


E    F 


determine  the  angle  COA,  after  which  the  desired  curve  AGE 
may  be  located. 
Draw   HOi  and  L0t  parallel  to  .E^aiid  NQ. 

EL  =  OiK  =  p. 


LOCATION.  109 

From  triangles  00,' Hand  00, L, 

(R  -\-  Ri}  cos  b  =  (R  +  Ri)  cos  a  —  p. 

.'.  cos  b  =  cos  a  -  RV  R (131) 

Angle  AOC  =  b  —  a. 

137.  To  Find  the  Radius  of  a  Curved  Track. 

Measure    any  chord   AB  =  21,   and  rnid-ordiuate   CE  =  M. 


FIG.  65. 
Then  in  the  right  triangle  OAE  (Fig.  65), 

R*-  (R  _  Jf  )*  = 


(132) 


CHAPTER  IV 


TRANSITION-CUR  VES. 


ARTICLE  11. — THEORY  OF  THE  TRANSITION-CURVE. 

138.  Elevation  of  Outer  Rail  on  Curves.— To  counteract  the 
effect  of  centrifugal  force  on  curves  the  outer  rail  must  be 
elevated  above  the  inner  one.  It  is  shown  in  mechanics  that  the 
centrifugal  force  is 


F  = 


32.16.ft' 


where  W  is  the  weight,  v  the  velocity  in  feet  per  second,  32.16 

an  average  value  of  the  acceleration  of  gravity  in  feet  per  second 

per  second,  and  R  the  radius  in  feet. 
In  Fig.  66  let  the  vertical  HL  represent  W,  the  horizontal  KI1 

the  centrifugal  force,  AB  the  plane  of  the  rails,  and  CB  —  e 
the  superelevation  of  outer  rail 
From  similar  triangles, 


Equate  this  value  of  F  to  that  given 
above  and  solve  for  e,  giving 


(133) 


32.16.ff' 


The  gauge  AB  should  be  greater  on  curves  than  on  tangents 
to  allow  for  flange  clearance  and  the  effect  of  a  rigid  wheel-base. 
AC  =  4.9  feet  is  about  the  right  value  for  the  horizontal  distance 
between  centers  of  rail-heads  for  standard  gauge.  In  formula 
(133)  v  is  in  feet  per  second,  but  the  train  velocity  is  usually  given 
in  miles  per  hour.  Let  V  =  velocity  in  miles  per  hour,  then  the 

110 


TRAKS1TIOK-CURVES.  Ill 

22 

velocity  in  feet  per  second  will  be  v  =  — -  V.     luserting   these 

•  15 

values  in  (183)  gives 

4.9  X  484 F3        F 
"g  =  82.16  xa25J8==8 

This  elevation  will  be  required  from  the  P.O.  to  the  P.T.,  but 
obviously  it  cannot  be  introduced  suddenly,  so  that  for  easy 
riding  the  rate  of  increase  of  e  should  be  uniform.  From  (134)  it 
is  seen  that  e  varies  inversely  with  7?,  which  requires  that  when 
e  =  0,  R  =  infinity.  Hence  It  must  decrease  from  infinity  to 
the  radius  of  the  circular  curve,  while  e  increases  from  0  to  its 
maximum  value. 

139.  The  True  Transition- curve  should  satisfy  formula  (134), 
but  so  far  no  such  curve  has  been  found  that  will  at  the  same 
time  admit  of  the  same  ease  of  location  as  the  simple  circular 
curve.      According  to  Raukine   the  first  use  of  any  other  than 
the  circular    curve  was   made  by  Gravatt   about  1828  or  1829, 
the  curve  employed  being  the  curve  of  sines.     Another  method 
described  by  Raukine   is  attributed   to   William   Fronde  about 
1842  ;    this  curve  was  worked  up  in  the  Engineering  News  by 
A.  M.  Wellington  in  1890.     Other  approximations  are  the  Hail- 
road  Spiral,  developed  by  W.  H.  Searles  in  1882,  and  the  cubic 
parabola,  described  by  C.  D.  Jameson  and  E.  W.  Crellin  in  the 
Railroad  and  Engineering  Journal,  1889. 

In  1880  Ellis  Holbrook  described  in  the  Railroad  Gazette  the 
true  transition-curve  applicable  to  small  angles  and  short  lengths 
of  the  curve.  In  1893  C.  L.  Crandall  published  formulae  and 
tables  applicable  to  large  central  angles  for  both  the  offset  and 
deflection  methods. 

140.  The  Notation  here  employed   will    be    explained   with 
reference  to  Fig.  67.      The  curve  CBB'C'  is  the  circular  curve 
offset  at  C  and  C'  from  the  tangents  by  the  amounts  CH  and  C'H'. 
AGE  and  B'G'A'  are  the  transition-curves.     A  is  the  P. T.  C., 
or  point  of  transition-curve,  C  the   P.O.,  B  the  P.C.,,  B'  the 
P.TC.,,  C'  the  P.T.,  and  A'  the  P. T.,.     The  co-ordinates  of  G 
are  .477  =  x',  HO  =  y';  of  C,  ;?''  and  77(7  =  F-   of  B,  AM  =  x, 
and  MB  —  #,.     The  length  of  curve  from  P.  T.  C.  to  any  point  P 
is  I,  and  the  whole  length  from  P.T.C.  to  P.  C.i  is  J,. 


1  12     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

141.  Equation     of    Transition -curve. — Since     the     rate     of 
change  of  e  must  be  uniform,  (134)  may  be  written 


(135) 


FIG.  67. 

in  which  k  is  the  rate  of  rise  of  outer  rail  along  curve,  and  p  the 
varying  radius  of  curvature.  From  the  calculus  pd<p  —  dl, 
whence 

dl 


P       dcf> 


(136) 


Insert  this  in  (135)  and  solve  for  d(p. 

d<p  =  -yjdl  =  2mldl (137) 

2m  is  dependent  upon  V  and  k,  and  is  constant  for  any  one 
curve. 
Integrating  (137), 

0  =  ml\ (138) 

the  constant  of  integration  being  zero,  for  I  is  zero  when  tp  is 
zero. 


TRANSITION-CURVES.  113 

From  the  elementary  triangle  drawn  at  the  point  P  of  transi* 
tion-curve,  Fig.  67,  dl  being  tangent, 


=  su 


Expanding  sin  0  by  trigonometry, 


in  which  3!  =  3x2Xl,  51  =  5x4x3x2x1,  etc. 
Substituting  for  0  its  value  from  (138), 


mH" 


Integrating, 


76600 


, 
+  -" 


But  ml*  —  <f>,  where  0  is  in  circular  measure.     To  obtain  0  in 
degrees,  0  =  0°^  =  57-3-     Inserting  this  in  (139), 


171.89      79  X  105      8151  X  108       153245  X  1012 

or  y  =  IC,      .......     (140) 

in  which  0  may  be  found  from  Table  XIV  with  0°  as  argument. 
Interpolation  must  be  resorted  to  for  values  of  <p  not  given  in  the 
table,  or  y  computed  by  the  formula. 

From  the  elementary  triangle  at  P,  Fig.  67 

dx 

~  =  cos  0. 

Expanding  by  trigonometry, 


114     A    FIELD-MANUAL.  FOR   RAILROAD    ENGINEERS. 

Substituting  ml"*  for  (/>  and  integrating, 

\ 


Replacing  ml*  by  0  reduced  to  degrees, 

-id      *&-+     ^—        _^!__          \ 

'  Y   32828  T  2328  X  106   33114  X  1010  "*"  '  7 

or     x  =  I  —  IE.  .......  (142) 

Ovaries  with  0°,  and  may  be  taken  from  Table  XIV  with  0° 
as  argument. 

142.  The  Transition-  curve  Angle  /j  is  the  value  0  assumes 
at  the  P.  a  i.     From  (138), 

/i  =  wJi2  .........     (143) 

From  (137)  and  (136), 


At  the  P.G.i  p  =  R  and  may  be  taken  equal  to  --,  so  that 


_  ft  __ 

%mli  ±r  ' 

whence 

1             D° 
m  =  2^=lT46W (144) 


This  value  of  m  in  (143)  gives 

J,         D0li 


(145) 


Reducing  this  to  circular  measure  by  writing  J^/,0^—  -~. 


gives 


143.  The  Coordinates  of  any  point  on  the  curve  are  given  by 
(140)  and  (142).     The  length  of  the  transition-curve  being  known 


UNIVjL 

J\J 

or  assumed,  ^i  and  Xi  (the  coordinates  of  the  P.C.\)  may  be 
found  from  these  equations  by  the  help  of  Table  XIV;  the 
coordinates  of  the  P.  (7,  (see  Fig.  67)  will  be 

F  =  yl  -  R(l  -  cos  /,)  =  y,  -  7?  vers  7i  ,   .     .     (147) 
x'  =  x,-RsinIi  ...........     (148) 

144.  Deflection-angles.—  With  the  transit  at  the  P.T.C.  (or 
P.T.i  in  backing  up)  the   tangent  of  deflection-angles  may  be 

found  from  the  relation  tan  d  =  -.     Dividing  (139)  by  (141), 


tan  d  =  -  -  +  .009523w8*6  -f  .000167m^10  +  ____     (149) 
o 

From  trigonometry  the  expansion  of  the  angle  in  terms  of  its 
tangent  is 

6  =  tan  d  —  i  tan3  d  -f  1  tan5  8  —  etc.       ...    (a) 
In  (149)  write  ml*1  =  0  and  substitute  in  (a)  : 


d  =  1  -  .00282303  -  .00006805 (150) 

9 

From  (138)  and  (143), 

*  =  H&  =  £=.nt 
Ii      mlS       IS 

in  which  --  =  n.     From  (ft),  0  =  /,TI-,  and  this  in  (150)  gives 

8  =  iv  -  .0028237W  -  .0000687, 5n>°.  .     .     .     (e) 

Both  d  and  7,  are  in  circular  measure  ;  to  reduce  to  degrees 
multiply  by'  — r— .  This  gives,  neglecting  terms  involving  higher 
powers  ot  7i  than  the  third, 

5°  =  ~^  -  .00000086  7x3?i6 (151) 

The  second  term  is  quite  small,  and  in  most  cases  may  be  en- 
tirely neglected  in  practice. 


116     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

With  the  instrument  at  auy  intermediate  point  x"y"  the  deflec- 
tion-angle for  any  point  xy,  measured  from  initial  tangent,  will  be 


tan  d  =  —,  =  iM»  +  ml"*  +  mil" 

...,  (152) 


in  which  powers  of  mP  higher  than  the  third  have  been  neglected. 
Substitute  the  value  of  tan  d  from  (152)  in  (a),  write  ml2  =  <p  = 
7,n2,  ml'"1  —  0"  =:  7jft"a,  by  (b),  and  reduce  circular  measure  to 
degrees,  giving 

S0  =  -L(W2  _j_  nn»  _|_  nnf^  _  a  small  correction.        (153) 
o 

For  instrument  at  P.  T.  C. ,  n"  =  0  ;  then  (153)  yields 

(#0°)  =  —  ri*  —  correction, 
or 

(do°)  =  *LA0  -  B0 (154) 

(154)  is  the  same  as  (151),  as  it  should  be. 

For    the    transit    at     the     quarter-point     of    transition-curve 

n"  =  j  =  y-1  =  ^;  then  (153)  yields 

7  ° 

(n*  -f-  T^  -\-  ^n)  —  correction, 
IP- 


V-t     ,  g    V" 

or 

T.° 

-*i (155) 


For  transit  at  mid-point  of  transition-curve  n"  —  \,  and,  from 
(153), 

(#£°)  =•  -^-(?i2  +  i  +  ?n)  —  correction, 
or 


TEA  NSITION-CURVES. 
For  transit  at  three-quarter  point  n"  —  f  and 

o\   —       l    (™1      I       9     _L    3*A 


117 


/   «>      O\  A       /       Q 

(5* )  =  •*••(*' 


or 


(157) 


For  transit  at  P. C.i  n"  =  1  and 

((V)  =  -^  (ft2  -f  1  +  n)  -  correction, 

o 


or 


(158 


With  the  transit  at  the  P.T.C.i  it  will  frequently  be  most  con- 
venient to  measure  the  deflections  from  the  tangent  to  the  circular 
curve  at  that  point.  Sometimes  this  will  also  be  the  case  for  the 
transit  at  the  P.  G. , . 

By  reference  to  Fig.  68  it  will  be  seen  that  for  the  transit  at  B 

A  c 

w r= 

P.T.C.  -l 

-    .*  -r^.6  * 


the  deflection  from  the  tangent  BG  which  serves  to  fix  any  point 
on  the  curve,  as  .6,  is  given  by  the  equation 


A  =    i    -     -«i  ~ 


or,  in  general, 


(159) 


118     A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

Table  XV  gives  the  values  of  A  and  B  for  the  five  positions  of 
instrument  for  which  equations  (154)  to  (159),  inclusive,  were  de- 
duced. The  value  of  A  must  be  multiplied  by  -jr- ,  but  B  is  taken 

o 

direct  from  the  table  in  thousandths  of  a  degree. 

If  deflection-angles  are  wanted  for  other  positions  of  the  instru- 
ment, or  for  other  points  on  the  curve,  they  may  be  computed 
from  equation  (153). 

145.  Tables. — Three  tables  are  given  for  use  with  transition- 
curves. 

Table  XIV  was  computed  for  use  with  formulas  (140)  and  (142) 
in  determining  C  and  E ;  <p  being  assumed  and  G  and  E  com- 
puted. 

Table  XV  gives  A  and  B  for  computing  the  deflection-angles 
by  (154),  (155),  (156),  (157),  (158),  and  (159)  for  20  equidistant 
stations  on  the  transition-curve.  For  points  not  given  in  the 
table  A  and  B  must  be  interpolated.  Linear  interpolation  will 
suffice  in  most  cases,  though  when/i0  is  quite  large  second  differ- 
ences may  be  preferable  for  A.  B  is  given  in  the  table  in  thou- 
sandths of  a  degree. 

Table  XVI  was  calculated  by  assuming  li  in  lengths  varying 
by  increments  of  20  feet,  then  computing  /i°  by  (146),  y\  by  (139), 
xl  by  (1.41),  F  by  (147),  and  x'  by  (148).  y,  and  x,  will  also  be 
given  more  directly  by  (140)  and  (142)  with  the  aid  of  Table  XIV. 

The  excess  in  length  of  transition-curve,  measured  from  P.  T.C. 
to  the  point  on  offset  at  P.O.,  over  x'  is  tabulated  as  e;  I'  is 
found  by  trial  such  that  when  inserted  in  (141)  or  (142)  the  same 
value  of  x'  will  be  obtained  as  in  (148).  This  may  be  done  by  as- 
suming I'  a  little  less  than  ^  ,  then  computing  x1.  More  than  two 

trials  will  rarely  be  needed  to  find  a  sufficiently  close  value  of  l'\ 
then  e  —  I'  —  x'.  y'  is  found  by  (139)  after  finding  I',  or  tp' 
may  be  found  from  (b)  of  144,  and  used  in  (140)  in  connection 
with  Table  XIV.  li  -  I'  is  the  length  from  G  (Fig.  67)  to  the 
P.O.j;  the  difference  in  length  between  this  and  the  length  of 
circular  curve  from  P.  C.  to  P.  C.i  is  tabulated  as  e' ;  that  is,  e'  = 
(j,  _  i')  _  arc.  Then  e  -\-  e'  =  li  —  (xf  -f  circular  arc). 

For  values  of  h  intermediate  between  those  given  in  the  table 
linear  interpolation  will  suffice,  though  second  differences  may 
be  used  for  F  &ud  y\  if  preferred. 


TRANSITION-CURVES. 


119 


146.  To  Unite  the  Two  Branches  of  a  Compound  Curve  by 
a  Transition-curve. 

The  same  objections  hold  to  compound  curves  as  to  simple 
curves  uniting  with  a  tangent;  i.e.,  where  there  is  a  sudden 
change  of  curvature  there  should  be  a  sudden  change  of  super- 
elevation of  outer  rail,  which  of  course  is  not  allowable.  Instead 
of  compounding  the  curves,  we  may  offset  them  at  the  P.  C.  C. 
and  unite  them  by  means  of  a  portion  of  a  transition-curve  tangent 
to  each  of  the  simple  curves. 

In  Fig.  69  AB  and  CELM&re  the  simple  curves  that  are  to  be 
united  by  the  transition-curve  ANE.  Extend  the  transition-curve 


to  G,  where  its  radius  of  curvature  becomes  infinite,  and  let  OS 
be  its  tangent.  Call  the  length  of  transition-curve  from  G  to  A 
li  ,  from  G  to  E  13 ,  and  from  E  to  A  I*.  E  and  A  are  points 
of  taugency  of  simple  and  transition  curves.  Then  It  •=  ll  —  13. 
The  coordinates  of  A  are  G8=  xlf  8A  —y^  ;  and  of  V (WV 
perpendicular  to  GS),  GW=x1t,  WV  =  F,;  of  E,  OP  =  ara  , 
EP  =  ya ;  of  L  (LH  perpendicular  to  GS),  GH  =  xs ',  HL  =  F3. 
Let  EC  =  Ft. 

The  radius  of  curvature  of  transition-curve  is  inversely  pro- 
portional to  its  length  from  G  ;  hence  the  curvature  is  propor- 
tional to  the  length  of  curve;  therefore  13  :  ^  =  D3  :  DI  ,  whence 


(160) 


120     A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 
Then         I,--    £  -  i,  =  Jl  -  jj\  =  li^jf-^'.   .    .     (161) 

By  (138)  or  (143),  ~  =  (*f)\ 

Equating  the  value  of  73  from  this  equation  to  that  resulting 
from  (146)  gives 

w <163> 

WV=  Fi  and  HL  =  F3  may  be  taken  from  Table  XVI  with 
li  and  la  as  arguments.  Then  Ol  W  —  ft  -f  Fi  03H  =  7?3  -j-  ^3. 
Draw  O,?7 parallel  to  #£,  then  OxT"=  IF//;  hence 

03T  =  (K3  +  Fs)  -  (ft!  +^), 
and  OiT  =  a*,'  —  ««'. 

Therefore 


^,  =   «,'  —  oV)  cosec  a  = 


The  lengths  of  yl#  and  CE  are 


m  .    .  (164) 


(165) 


(166) 

CE=        ~D*   .........     (167) 

The  excess  of  transition-curve  length  over  AB  -|-  CE  is 


TRANSITION-CURVES.  121 

If  AB  and   CE  are  quite  sharp,  we  must  take  account  of  the 
arc  excess,  so  that  we  have  then 


arc  excess 


The  arc  excess  may  be  taken  from  the  second  column  of 
Table  IV,  which  gives  the  arc  length  for  one  station;  this  multi- 
plied by  the  number  of  stations  gives  the  curve  length,  which 
may  replace  the  values  within  the  brackets  in  (168'). 

147.  Length  of  Transition-curve  to  be  Taken. — In  practice 
the  rate  of  change  of  superelevation  of  outer  rail  may  vary  from 

TO-  to  J.    Call  the  rate  k  ;  then  evidently  kl,  must  equal  the 


superelevation  of  outer  rail  for  circular  curve  ;  or,  by  (135), 

F2 


Writing  R  =  — =--,  and  solving  for 


17190&  ' 


For  k  = 


For  k  = 


1200' 

1 
600' 


400' 


(169) 
(169') 


'tl  =  0.035  F2D (169") 

I,  =  0.023  F2Z> (169'") 


The  following  table  gives  values    of  Z,  in  feet  per  degree  of 
circular  curve  for  a  few  values  of  Fand  k. 


fc 

30  Miles 
per  Hour. 

35  Miles 
per  Hour 

40  Miles 
per  Hour. 

45  Miles 
per  Hour. 

50  Miles 
per  Hour. 

55  Miles 
per  Hour. 

1 

1200 

63 

86 

112 

142 

175 

212 

1 
600 

32 

43 

56 

71 

83 

106 

1 

400 

21 

29 

37 

47 

58 

71 

122     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

When  only  a  short  tangent  intervenes  between  two  curves 
shorter  transition-curves  must  be  taken,  requiring  larger  values 
of  #,  so  that  overlapping  may  be  prevented. 

For  illustration  suppose  a  5°  curve  to  be  eased  off  with  a  tran- 
sition-curve, the  highest  train-speed  being  45  miles  per  hour  and 

*  =  *e*t     Bv  the  table  tbe  value  of  ll  wil1  be  71  X  5  =  355  feet, 
bU(J 

so  that  we  should  probably  take  a  360-ft.  transition-curve,  re- 
quiring an  offset  of  4.7  feet  by  Table  XVI. 

ARTICLE  12  —  FIELD-WORK. 

A      Field  Formulas. 

148.  For  the  cases  most  frequently  presenting  themselves  in 
practice  the  foregoing  formulas  may  be  simplified  so  as  to  admit 
of  the  rapid  location  of  points  on  the  transition-curve  with  all  the 
accuracy  needed  on  location,  though  it  is  best  to  use  the  exact 
formulas  and  tables  in  setting  track-centers  on  the  finished  road- 
bed.    When  the  transition-curve  angle  is  quite  large  it  will  be 
better  to  use  the  accurate  methods  on  location  also,  but  for  the 
more  common  cases  the  following  formulas  will  answer. 

149.  Simplified    Formulas.  —In   (139)  and   (140)   neglect,    as 
small,  all  the  terms  following  the  first,  giving 


y  =         =     i  =  .005818Z00.  .  (170) 

o          o 


In  (141)  and  (142)  retain  only  the  first  two  terms 


in  which  the  last  term  is  small  for  short  transition-curves  and 
may  often  be  neglected,  x  being  taken  equal  to  I. 
The  values  of  m  and  /i  remain  as  before  : 

1  D  (144) 


1146W, 


TRANSITIOK-CURVES.  123 

In  (147)  expand  cos  7i ,  giving 


But  R  =  — r,  by  (145).     Substitute  this  for  R  and  neglect  all 
but  the  first  two  terms  : 


But  lili  =  3^i  ,  by  (170),  since  0  =  /i  and  y  =  yl  when  l  =  li\ 
hence 

^=yi-tti  =  iyi       .....     (172) 

Likewise  expanding  sin  /i  in  (148), 


and  writing  .R  =  ^  as  above, 
-^^1 

^i   .   US 
X='Xl-2+-&-'- 

But  «»  =  h  -   ^^j2,  by  (171). 

...  cc'  =  ,1_^  +  ^!-^|(nearly).  .         (173> 
By.  (170), 


In  (154),  (155),  (156),  (157),  (158),  and  (159)  neglect  the  correc- 
tion; then 

(«V)  =        4o  .......     (175) 


124     A   FIELD-MAHZJAL   FOR   RAILROAD    ENGIHEERS. 


(179) 
(180) 


150.  Offsets.  —  Formula  (170)  shows  that  offsets  from  transition- 
curve  to  tangent  vary  as  the  cube  of  the  distance  from  the  P.  T.C., 
and  it  can  be  shown  that  offsets  from  the  circular  curve  to  tran- 
sition-curve follow  the  same  law,  reckoning  from  the  P.G.i. 

Formula  (36)  may  be  written 


(a) 


in  which  D  is  the  degree  of  curve  if  offset  is  from  tangent,  and  the 
difference  of  degrees  if  offset  is  between  two  curves  having  a  com- 
mon point  of  tangeucy,  I  being  reckoned  from  the  tangent-point. 
From  (136)  and  (137), 

dL  _L 

dp  ~~  P  ~  2ml' 

and  the  degree  of  transition-curve  at  any  point  is 


Dt  =  —  =  11460m*  =  el  .....     (181) 

Formula  (181)  shows  that  the  degree  of  curvature  of  transition- 
curve  at  any  point  is  a  function  of  its  length.  If  the  D  in  (a)  is 
the  difference  between  degrees  of  circular  and  transition  curves,  it 
will  equal  D\  —  Dt  ,  which  is  also  a  function  of  the  length;  so 
in  (a)  write  D  =f(l),  giving 


which  shows  that  the  offset  between  circular  and  transition  curves 
varies  as  the  cube  of  the  distance  from  P.C.\.  The  offset  at  the 
P.  C.  is.  known,  being  half  of  F,  and  may  therefore  be  found  for 


TRANSITION-CURVES.  125 

other  points  ;  thus  midway  between  P.  C.  and  P.C.i  it  will  be  one 
eighth  of  its  value  at  P.C.,  or 


151.  Compound  Curves.  —  By  trial  it  has  been  found  that  et 
-see  formulas  (168)  and  (168')—  equals  e  +  e'  from  Table  XVI 
when  the  table  is  entered  with  D  =  Di  —  D3  and  I*  as  arguments, 
up  to  about  Dili  —  8000,  which  covers  all  cases  in  ordinary  rail- 
road practice  ;  so  we  write 


(183) 


The  distance  AB  on  sharper  curve  is  found  by  trial  to  equal 
fa  up  to  about  Dili  =  4000,  which  answers  for  the  ordinary  cases 
arising  in  practice.  When  Dili  is  greater  than  this  AB  (of  Fig. 
69)  must  be  found  from  (166). 

The  point  N  can  be  taken  midway  between  0  and  B,  for  the 
radius  of  transition-curve  decreases  uniformly  from  -JB3  to  JKi  and 
may  be  here  taken  as  their  mean  ;  hence  the  offset  B  N  =  NC  = 
%Fz*  Other  offsets  may  be  found  from  the  relation  given  by  (182). 
Thus  the  offset  midway  between  A  and  B  will  be 


2   \faj        16 
Other  offsets  may  be  obtained  if  desired. 

B.  Setting  Out   Transition-curves. 

152.  In  first  locating  the  line  it  will  be  sufficient  to  simply 
offset  the  curve  at  the  P.  C.  the  amount  required  for  the  transition- 
curve  ;  then,  with  the  transit  over  this  point,  bring  the  telescope 
parallel  to  the  tangent  from  which  offset  was  taken  and  run  the 
circular  curve  to  the  P.T.,  where  another  offset  is  made  and  a 
tangent  parallel  to  the  terminal  tangent  of  circular  curve  can  be 
run  out.     The  amount  to  offset  will  be  governed  by  the  length  of 
transition-curve,  or  if  the  offset  is  fixed  it  governs  the  length  of 
curve.     Either  li  or  F  being  given,  the  other  may  be  taken  from 
Table  XVI. 

153.  Location  by  Offsets, — If  the  offset  is  given,  ^  can  be  taken 
from  Table  XVI,  interpolating  if  necessary.     At  the  P.  C,  bisect 
the  offset,  and  set  a  stake  at  that  point ;  then  measure  back  along 
tangent  the  distance  x' ,  which  for  most  cases  may  be  taken  as  fa 


126     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

(see  formula  (173)  ),  and  set  a  stake,  marking  it  P.  T.  C.  From  the 
P.  C.  measure  forward  around  the  circular  curve  a  distance  equal  to 

~,  which  approximately  equals  Ui.     Set  a  stake  marked  P.  C.,. 

At  the  quarter-point  offset  from  tangent  an  amount  equal  to  TV^> 
for,  by  (170),  the  offsets  are  proportional  to  the  cube  of  the  dis- 
tance from  P.T.C.,  so  that 

=  I 

16   ' 

At  the  three-quarter  point  offset  the  same  amount  from  circular 
curve.  If  the  transition-curve  is  not  over  400  feet  long,  these  are 
all  the  points  need  ;  if  longer,  other  offsets  are  similarly  found 

EXAMPLE.—  At  sta.  412  an  offset  of  4.2  feet  was  made  from  a 
tangent  to  a  5°  curve.  Required  the  data  for  a  transition-curve 
to  connect  tangent  and  circular  curve. 

By  Table  XVI  it  is  seen  that  a  340-ft.  transition-curve  is  re- 
quired. From  the  table  it  is  seen  that  x'  =  169.9  ft.,  7,°  =  8.5°, 
and  excess  of  curve  over  tangent  is  .02  ft.,  which  we  neglect  as 
small.  Drive  a  stake  2.1  ft.  from  offset  hub  and  mark  it  412  ; 
measure  back  along  tangent  169.9  ft.  to  410  +  30.1,  and  drive  a 
stake  marked  P.T.G.  Measure  forward  around  circular  curve 

Q  t\ 

—  =  1.70  chains  =  170  ft.,  and  set  a  stake  marked  P.C.i  at  sta. 
5 

413  +  70.1 

The  approximate  offsets  are  : 

4  2 

At  mid-point,  sta.   412,  t  =  -  =2.1 


'  one-eighth  points,  stas.  *|!J  +  ^-6  )  t  =  21  x(i)3=  0.033 
'  quarter-points,  stas.  ^  +  gj?'  j  j-  ,  t  =  0.033x23  =  0.26 
"  three-eighths  points,  stas.^J  ^  ^'^|  ,  t  =  0.033  X  33  =  0.89 


Stakes  at  the'  one-eighth  and  three-eighths  points  were  not 
needed,  but  were  worked  out  for  illustration. 

154.  Location  by  Deflections.  —  The  number  of  chord-lengths 
being  taken  as  an  aliquot  part  of  20,  the  deflection-angles  for  the 


TRANSITION-CURVES.  127 

transit  at  any  one  of  five  positions  may  be  taken  from  Table  XV  by 
multiplying  the  tabular  values  of  A  by  -^-,  7i°  being  found  from 

Table  XVI  or  formula  (146).  If  the  number  of  chords  is  not  an 
aliquot  part  of  20,  or  if  the  transit  is  at  some  point  other  than  one 
of  the  five  for  which  Table  XV  was  calculated,  then  the  deflec- 
tion-angles must  be  computed  by  (153).  The  curve  is  then  run 
out  in  the  usual  way. 

When  /i  is  not  more  than  15  or  20  degrees  the  curve  may  be 
run  from  the  P.T.C.  or  P.T.C.i  by  neglecting  the  correction  B  as 
small.  Even  when  /i  is  greater  than  20°  the  correction  may  be 
neglected,  provided  half  the  transition-curve  is  run  from  the 
P.T.C.  and  the  remainder  with  the  transit  at  the  mid-point,  the 
telescope  being  first  placed  parallel  to  original  tangent. 

EXAMPLE. — Take  the  example  of  the  last  section:  Z,  =  340  ft., 

F  =  4.2  ft.,  D  =  5°.    By  formula  (146),  7t  =  -       —  =  8.5°,  the 

/&00 

same  as  given  by  Table  XVI.    Then  ~  =  2.833°.    Divide  ^  into 

o 

5  parts  of  68  ft.  each,  which  will  be  the  chord-length  to  be  used. 
From  Table  XV  for  transit  at  P.  T.  C.  the  deflections  will  be  : 

For  sta.  410  +  30.1,  P.T.C.,  (<V)o  =  0. 

"      "    410  +  98.1,  (<V)-2  =  2-833  X  .04  =  0.1133  =  0°    6.8'. 
"      "    411  -f-  66.1,  (<V).4  =  3-833  X  .16  =  0.4533  =  0°  27.2'. 
"      "    412  +  84.1,  (<V).6  =  2.833  X  .36  =  1°    1.2'. 
«      «    413  _j_  02.1,  (<V).8  =  2.833  x  .64  =  1°  48.8' 
"      "    413  -f  70.1,  (<V)i    =  2.833  X  1      =2°  50'. 

Having  set  out  the  transition-curve,  move  to  P.  C.i  at  sta.  413  -f- 
70.1,  backsight  to  P.T.C.i ,  and  deflect  /,°  —  (d0°),  =  8°  30'  - 
2°  50  =  5°  40',  and  run  out  the  circular  curve  to  the  P.I.O.i, 
which  suppose  to  fall  at  sta.  420.  Set  the  transit  at  this  point,  and 
cause  the  vernier  to  read  zero  when  the  telescope  is  in  tangent  to 
circular  curve.  The  deflections  taken  from  Table  XV  will  now  be : 

For  sta.  420  +  68,  (S0°\6  =  2.833  X  .56  =  1°  35.2'. 
"  "  421  -f  36,  (<V).6  =  2.833  X  1.04  =  2°  56.8'. 
"  "  422  +  04,  (Sc°).4  =  2.833  X  1.44  =  4°  4.8'. 
"  "  422  -f  72,  (Sc°).2  =  2.833  X  1.76  =  4°  59.2'. 
"  «  423  -f  40,  (<V)0  =  2.833  X  2  -=5°  40'. 


128     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


Set  transit  at  423  +  40,  the  P.77., ,  backsight  to  420  and  deflect 
8°  30'  -  5°  40'  =  2°  50',  when  the  telescope  will  be  in  tangent. 

155.  Form  of  Transit  Notes.— The  following  will  illustrate 
a  form  of  notes  that  will  be  found  to  answer. 

Let  the  P.O.  of  a  4°  aurve  be  at  sta.  160  +  50,  and  a  200-ft. 
transition-curve  be  employed.  Let  the  intersection-angle  I  be 
20°.  By  Table  XVI,  F  =  1.16  ft.,  J,°  =  4°,  x  =  100  ft.,  so  that 
P.T.C.  is  at  159  +  50.  Take  four  50-ft.  stations  on  transition- 
curve  and  determine  the  deflection-angles  as  in  the  last  section. 


Sta. 

Deflec- 
tion- 
angle. 

Central 
angle. 

Calcu- 
lated 
Course. 

Mag- 
netic 
Course. 

Remarks. 

167 

+50  v, 

P.T.J. 

2°  40' 

4°0' 

166 

2°  15' 

+50 

1°40' 

165 

0°  55' 

+500 

P.T.C.! 

"6°'6'" 

12°  0' 

164 

5°0' 

163 

3°  0' 

162 

1°0' 

+500 

P.C.,,4  C.L. 

'  i"°"  20'  ' 

4°0' 

Set  ver.  at  2°  40', 

161 

0°45' 

B.S.  to  159  +  50,  and 

+50 

0°20' 

deflect  to  0°.     Run 

160 

0°5' 

circular  curve. 

+50© 

P.T.C. 

0°0' 

lt  =  200.  F  =  1.16. 

159 

158 

The  length  of  circular  curve  was 
the  central   angle  was  12°.     With 


20  -  2  X  4 


=  3  stations,  since 


4 

transit  at  161  +  50  set  the 
vernier  to  2°  40',  backsight  to  159  -j-  50,  and  deflect  into  taugent 
with  the  vernier  reading  zero.  With  the  transit  at  164  -f-  50 
cause  the  vernier  to  read  zero  when  the  transit  is  in  the  tangent  to 
circular  curve,  and  run  the  Jast  transition-curve  by  deflections 
from  this  tangent.  With  the  transit  at  166  +  50  backsight  to 
164  -f  50  and  deflect  4"  -  2°  40'  =  1°20',  when  the  telescope  will 
be  in  tangent  and  the  line  may  be  continued. 


TRANSITION-CURVES. 


129 


ARTICLE  13.     TRANSITION-CURVE  PROBLEMS. 

156.  To  Find  the  Tangent  Distance  and  External  when 
Transition-curves  are  Employed,  Offsets  Equal. 

In  Fig.  70  let  AB  be  the  circular  curve,  EF  and  OH  the  tran- 
sition-curves. Let  EK  —  HK  =  Tl  be  the  tangent  distances,  and 
NK—  El  the  external  required.  Let  LK  =  T '.  Draw  PV  per- 


pendicular  to   LK;    then  in  triangle   PVK,    VK  =  F  tan  f/; 
LK  =  AP  -}-  'VK  is  now  known,  or 


T'  =  T  +  Finn^I.  ..........     (184) 

Hence 

T,.  =x'+  T'  =  x'  +  r+^tany.  .     .     .     .     (185) 

'In  triangle  PVK,    PK  =  Fsec  |/,  so  that,  letting  PN  =  E, 

E,  =  E  +  F  sec  |7.     .      .     ,     .     .....     (186) 

EXAMPLE.—  Two  tangents  intersect  at  sta.  91  -f-  37  8;  required 
the  tangent  and  external  when  F  —  2.62  ft.,  /  =  26°  30',  D  =  4°. 

By  Table  XVI,  I,  =  300,  *'  =  149.9.  From  Table  IX,  T-  ^^ 
=  337.3.     Then,  by  (185), 

Ti  =  149.9  -f  337.3  -f-  2.62  tan  13°  15'  -  487.8  ft. 

The  station  number  of  P.T.C.  will  now  be  91.378  -  4.878  = 
86       50. 


130     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 
By  (186), 


+  2.62  sec  13°  15'  =  41.87  ft. 


Table  XVI  gives  ^1°  =  6°;  hence  the  circular  curve  will  cover 
26°  30'  -  2x6°=  14°  30',  or  3.625  stations,  so  that  the  number 
of  the  P.T.,  will  be  86.50  +  (2  X  3.00  -f  3.625)  =  96  +  12.5. 

157.   Tangent  Distance,  Offsets  Unequal. 

In  Fig.  71,  0.  2?,  and  K  do  not  lie  in  the  same  straight  line. 


_— T-l — J 


Draw  PS  perpendicular  to  NB,  PQ  perpendicular  to  LK.     Let 
LA  =  F,  MB  =  F'. 


T '  =  LK  =  AN  -f  NP  -  KQ, 

T'  —  T  +  F'  cosec  /—  ^cot  /; (187) 

T_x  ±J"  =x'  +  T+  Flcpsfin  T-  Jfrnt.  r:    (188) 


or 


T"  =  MK  =  T  -  F'  cot  /  +  F  cosec  7  ;    .     .     .     (189) 
r2  =  x"  +  r  —  ff"  cot  I  +  F  cosecJL^^-  .     .    (190) 

EXAMPLE — Two  tangents  intersect  at  sta.  820  and  arc  to  be 
united  by  a  6°  curve  having  F  -.  1.75,  F'  —  2.95,  and  /=  31n  48'. 


By  Table  IX,      T  - 


=  272.05  ft. 


By  Table  XVI,  L,  -  200,  19  =  260,  x'  =  100,  x"  =  129  9. 


TKANSITION-CUKVES. 


131 


By  (188), 
Ti  =  100  +  272.05  +  2.95xcosec31°48'  —  1.75  cot  31°  48'  =  874.8. 

By  (190), 
r2  =  129.9  +  272.05  — 2. 95  cot  31°  48' +  1.75  cosec  31°  48' =400. 5. 

158.   To    Insert    Transition-curves    without   Changing    the 
Position  of  the  Vertex,  B 

In  Fig.  72,  ABC  is  the  located  curve,  FQHK  the  curve  after 


F       A 


N  M 


FIG.  72. 

inserting  transition-curve.  The  radius  of  the  circular  portion  has 
been  changed  from  R  to  R'  in  order  to  make  room  for  the  offset 
PS  =  F.  BM  =  E  is  the  external  to  located  curve,  BL  =  E1 
the  external  to  circular  curve  having  radius  R'  and  central  angle 
/.  In  the  triangle  LNM,  LM  =  LN  sec  \I  =  F  sec  -|/;  hence 


E'  =  E  -  Fscc±l. 


(191) 


E  may  be  found  by  (24)  or  by  means  of  Table  IX  ;  then  E' 
becomes  known,  and  from  the  same  table  Df  is  found  by  dividing 
the  tabular  E  by  E'.  D'  will  be  larger  than  D. 

It  is  sometimes  more  convenient  to  assume  D'  and  calculate  E' 
in  the  same  manner  as  E;  then,  from  (191), 


F=(E-  E'}  cos 


(192) 


If  this  value  of  Fis  too  large  or  too  small  for  the  conditions  of 
the  problem,  a  new  D'  can  be  assumed  and  2^  recomputed. 


132     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

EXAMPLE.—  The  P.  C.  of  a  5°  curve  is  at  sta.  182,  and  angle 
/  =  40°.  Compute  the  data  for  a  new  curve  to  allow  for  a 
transition-curve  with  1.5  ft.  offset. 

From  Table  IX,  E,  =  367.7  for  /  =  40°  ;  therefore 

E  =  ^1  =  73.54,     ^sec  20°  =  1.5  X  1.0642  =  1.6  ; 
o 

then,  by  (191), 

E'  =  73.54  -  1.60  =  71.94, 
and 

D'  =  |^  =  5.1113°  =  5°  6.678',  say  5°  7'. 
By  Table  XVI,  for  I,  =  200,  D  =  5°  7' 


—  1.45  _}_   L(1.75  -  1.45)  =  1.485. 


For  J,  =  220, 


V-  1.76  +  6^(2.11  -  1.76)  =  1.842. 
Then  for  F  =  1.5,  D  =  5°  7', 

I,  =  200  +  20  /^"l1'/^  =  200-8  and  *'  =  100-4- 

™- 

The  central  angle  for  circular  portion  of  curve  is  40  —  2  X  5.13 
=  29.74°,  equivalent  to  581.2  feet  around  curve. 

In  Fig.  72,  B  is  at  sta.  186  on  the  5°  curve,  and  arc  BG  —  290.6 
ft.  on  the  5°  7'  curve.  The  P.C.i  is  at  186  —  2.906  =  sta.  183  -f 
09.4,  the  P.T.C.  at  183.094-2.008  =  sta.  181  +  08.6,  the 
P.  T.C.i  at  188  +  90.6,  and  the  P.T.i  at  190  +  91.4. 

Had  D'  been  assumed  equal  to  5°  6'  or  5.1°  to  begin  with, 

we  should  have  had  E'  =  -^-  =  72.10  ;  then,  by  (192), 
5.  1 

F=  1.44  X  .93969  =  1.35ft. 
li  may  be  found  by  interpolation  from  Table  XVI  as  above. 


TRANSITION-CURVES. 


133 


159.  To  Insert  Transition-curves  on  an  Existing  Road-bed 
with  the  Least  Deviation  from  Old  Track. 

To  satisfy  this  condition  the  new  track  should  pass  about  as 
far  outside  the  old  at  the  vertex  as  it  does  inside  at  the  original 


P.O.;  that  is,  about  — . 


We  shall  now  have 


(193) 


The  remainder  of  the  problem  may  be  solved  by  158. 

Transition-curves  may  be  inserted  in  old  track  by  shifting  to 
suit  the  existing  road-bed,  thus  adding  materially  to  the  safety 
and  easy  riding  of  cars. 

160.  To  Insert  Transition -curves  at  the  Ends  of  a  Long 
Circular  Curve  without  Moving  the  Central  Portion. 

In  Fig.  73,  AC  is  the  circular  curve.  In  order  to  make  room 
for  the  offset  F  the  ends  must  be  sharpened  by  compounding. 
Let  C  be  the  point  of  compounding,  R  the  radius  of  the  branch 
ON,  HN  =  KB  =  F.  Let  BEG  be  the  transition-curve  ;  the 


B 


L 
0 
FIG.  73. 


closer  O  comes  to  C  the  better,  provided  the  change  in  radius  at 
C  is  kept  within  certain  limits.  The  difference  in  degrees  between 
the  original  and  the  sharpened  curve  should  never  exceed  2°  and 
may  usually  be  kept  in  the  neighborhood  of  1°. 

FIRST  METHOD. — Having  decided  upon  the  value  of  F,  assume 
R'  so  that  D'  —  D  is  not  greater  than  2°.      Draw   O'L  parallel 

to  BH-  00'  =  R  -  It',  and  cos  /'  =         ,  Or 


134     A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


• 

This  is  the  same  as  (69)  in  122.  /'  being  known,  set  the  transit 
at  C,  run  out  the  curve  CN,  and  insert  transition-curve  in  the 
usual  way. 

If  /'  had  been  assumed  in  the  beginning,  R'  could  be  found 
from  (194). 

SECOND  METHOD.  —  When  the  circular  curve  is  flat,  and  short 
transition-curves  are  employed,  we  may  compound  the  transition- 
curve  with  the  circular  at  the  P.C.i,  taking  care  that  the  differ- 
ence of  curvatures  is  not  greater  than  1°  or  2°. 

Assume  the  position  of  the  P.  C.  i  from  100  to  200  feet  from 
the  P.  C.;  measure  the  perpendicular  let  fall  from  the  P.  C.i 
upon  the  tangent  at  the  P.G.  produced;  this  will  be  1/1.  The 
central  angle  I\  can  be  calculated,  knowing  the  length  of 
circular  curve  from  the  P.O.  to  the  assumed  P.C.i,  or  the 
angle  between  tangents  may  be  measured  with  the  transit. 
The  coefficients  C  and  E  of  (140)  and  (142)  may  be  found 
from  Table  XV  with  It  =  <p  as  argument  ;  then,  from  (140)  and 
(142), 


aj,  =  Z,(l  -  E)  .......     (196) 

Measure  back  from  the  foot  of  the  perpendicular  let  fall 
from  the  P.C.\  a  distance  #1  along  tangent,  and  set  the  P.T.G. 

Intermediate  points  can  be  located,  if  needed,  by  offsets 
from  tangent,  computed  by  (140)  or  (170)  ;  thus  at  the-  mid- 
point the  offset  is  ^yi. 

THIRD  METHOD.  —  From  formula  (170), 


and,  from  (36), 


TRANSITION-CURVES  135 

But  nD  =  L°;  hence 

Zi  =  150ft  ;    .......     (197) 

and  as  100/i  is  the  length  of  circular  curve  from  P.  C.  to  P.O.i, 
li  is  once  and  a  half  as  great. 
From  (146), 


_  2007,°  _  2007,  °  _  _      ^ 

~~  r  "^"      ' 


From  this  equation  it  is  seen  that  if  the  break  in  curvatures  is 
limited  to  2°,  this  method  is  admissible  up  to  D  =  6°,  independent 
of  the  length  of  transition-curve. 

EXAMPLE.  —  A  4°  curve  is  to  have  transition-curves  inserted  at 
each  end;  compute  the  necessary  data. 

BY  FIRST  METHOD  —  Assume  a  1.45-ft,  offset,  and  the  curva- 
ture to  be  changed  from  4°  to  5°  by  compounding.  In  Table  I 
find  R  =  1432.7,  R'  =  1146.3;  then,  by  (194), 

cos  7'  =  1  -  i       =  .99494  =  cos  5°  46'. 


K  '7C'7 

The  length  of  5°  curve  is  -—  —  =  1.153  stations,  and,  by  Table 
5 

XVI,  /r  =  5°,  so  that  the  P.  (7.,  will  fall  15.3  ft.  back  of  the 

P.C.C.,  while  the  P.O.  will  be  moved  forward  t^  -  1.153  = 

4 

.289  stations  or  28.9  ft.  ;  the  P.T.C.  being,  by  Table  XVI,  100  ft. 
back  of  the  new  P.O.  will  fall  100  -  28.9  =  71.1  ft.  back  of  old 
P.  C.  The  transition-  curve  may  now  be  located  in  the  usual 
manner. 

BY  SECOND  METHOD.  —  Assume  the  P.C.i  to  fall  150  ft.  from 
the  P.O.,  making  7,°  =  1.5  X  4  =  6°.  From  Table  XV,  C  = 
.03488,  and,  by  (36),  y,  =  |(1.5)a  X  4  =  7.875  ft. 

By  (195), 

7'875 


.03488 
Now,  by  (146), 

6  -.=  ?  X 


200 


130     A    FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

from  which  D'  =  5.314°  =  5°  18.8  ,  which  differs  less  than  2° 
from  D 
By  (196), 

xl  =  225.8(1  -  .0011)  =  225.6  ft. 

To  find  the  position  of  P.T.C.  with  reference  to  the  old  P. C. 
consider  that  the  distance  from  P.  C.  to  foot  of  perpendicular  from 
the  P.C.i  is  half  the  chord  for  angle  2Ii ,  and  can  be  taken  from 

1       1 197  Q 
Table  IX,  being  equal  to  -  X  —^-  =  149.7.   Then  225.6  -  149.7 

=  75.9  feet  is  the  distance  from  old  P. C.  back  to  P.T.C. 

BY  THIRD  METHOD. — Assume  the  P.G.i  to  be  150  ft.  from  the 
old  P. C.;  then,  by  (197),  J,  =  225  ft.,  and,  by  (198),  the  curvature 
of  transition-curve  at  the  •  P.C.i  is  |  X  4°  =  5°  20',  giving  almost 
the  same  results  as  by  the  second  method.  Had  we  taken  the 
P. C.,  160  ft.  from  P.O.  we  should  have  had  J,  =  240,  D'  =  5°  20'; 
xl  —  239.7,  by  interpolation  from  Table  XVI;  the  length  along 
tangent  from  P.O.  to  foot  of  perpendicular  from  P.C.i  159.9  ft., 
and  therefore  239.7  -  159.9  =  79.8  ft.  as  the  distance  from  P. C. 
to  P.T.C. 

161.  To  Insert  Transition-curves  at  the '  P.  C.  and  P.  C.  C.  of 
a  Compound  Curve  by  Changing  the  Curvatures  of  the  First 
Branch. 

In  Fig.  74  let  ABV  be  the  located  curve  compounding  at  B* 
Two  cases  occur. 

FIRST  CASE. — Second  brancJi  having  shorter  radius. 

The  offset  at  P.C.G.  must  be  to  outside  of  located  curves;  let  it 
be  EB  =  7^2  in  the  figure.  Let  CP  =  F  be  known  or  assumed. 

Draw  the  tangent  BO,  and  draw  EH  parallel  thereto.  Let  CE 
be  the  changed  curve,  and  CQ  parallel  to  tangent  .477.  Angle  / 
may  be  computed  from  the  known  station  numbers  of  A  and  7?, 
or  may  be  measured  on  the  ground.  The  new  tangent  distance  is 
EQ  =  BO  -  OK  —  HQ  (or  LS).  From  the  right  triangle  QHK, 
OK  =  HKtw  GHK  =  FS  cot  7. 

Similarly,  L8  =  L  W cosec  I=F cosec  /.     Therefore 

T'  =  EQ  =  T  -  F*  cot  7  -  F  cosec  /.     .     .     (199) 

T  can  be  found  from  Table  IX  or  formula  (14);  then  T'  is 
known  from  (199).  The  degree  of  new  curve,  7)',  may  now  be 
found  by  means  of  Table  IX,  or  from  Table  I  by  first  finding 


TRAKSITION-CURVES. 


137 


R'  by  (15)      The  transition  curve  at  the  P.C.C.  may  be  located  by 
146  and  151,  while  that  at  the  P.O.  may  be  located  either  by 
offsets  or  deflections. 
SECOND  CASE. — Second  brancli  having  longer  radius. 


FIG.  74. 


In  this  case  the  offset  must  be  to  inside  of  curve,  and  NS  is  the 
tangent  required.     From  the  figure,  letting  NB  =  F*,  1TS'=  T', 


T  =  T+F*  cot/  -  F  cosed.  .    . 


(200) 


The  remainder  of  the  solution  is  the  same  as  for  first  case. 

EXAMPLE. — A  5°  curve  compounds  at  sta.  280  with  a  9°  curve; 
the  P.C.  is  at  sta.  272.  Required  the  change  in  curvature  of 
first  branch  for  an  offset  of  1.50  ft.  at  P.C.  C.  and  2.00  ft.  at  P.C. 

Here  /  =  8  X  5°  -=  40°,  and,  by  (199), 


T  =  417.1 
2085.5 


D'  = 


412.2 


15  X  1.19175  -  2  X  1.5557  =  412.2 
=  5.06°  -  5°  3.6'. 


Z>2  -  Hf  =  9  —  5.06  =  3.94°  is  the  difference  in  curvatures  of 
the  two  branches  of  the  altered  curve.  Entering  Table  XVI 
with  this  value  for  D  and  F  =  1.5  ft.,  we  find  : 


138     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


for  lt  =  220, 
"   I,  =  240, 


F  =  1.06  -f  .94(1.41  -  1.06)  =  1.39; 
F  =  1.2G  +  .94(1.67  -  1.26)  =  1.65. 


.  •.  f  or  F  =  1 . 5,  D  =  3. 94° ,     Z,  =  220  -4-  20~~  — j  L  =  228. 5 

Bisect  the  offset  at  P. C.  C.,  measure  114.25  ft.  along  each  curve, 
and  set  the  ends  of  transition-curve.  Midway  between  these 
points  and  P.C.C.  offset  TJff  X  1.5  =  0.1  ft.;  these  are  all  the 
points  needed. 

The  length  of  transition-curve  at  P. C.  may  be  found  in  like 
manner,  taking  D  =  5.06°  and  F  =  2.0  ft.  as  arguments  in 
interpolating  in  Table  XVI. 

162.  To  Insert  Transition-curves  at  the  Ends  of  Two  Circu- 
lar Curves  of  Contrary  Flexure  united  by  a  Common  Tangent. 

In  Fig.  75  let  the  located  line  be  ABCE;  the  tangent  BC  must 
be  shifted  outward  at  B  and  C  to  the  position  HO ,  the  relative 
size  of  offsets  being  determined  by  the  nature  of  the  ground. 
The  points  B'  and  C'  at  which  the  tangents  to  circular  curves 


Fro.  75. 

will  be  parallel  to  TIG  will  each  move  towards  S  a  distance  due 
to  the  increase  of  central  angle,  which  increase  equals  BSII  =  a, 
for  which  we  have 


BIT-}- CO 
taa  a  =  — -— . 


Let  the  offset  at  B'  be  F,  and  at  C'  ,  F'.     Then 


F  = 
F'  = 


BH)  cos  a  -  Jt, 
CG)  cos  a  —  A". 


(201) 


(202) 
(203; 


TRANSITION-CURVES.  139 

F  and  F',  being  now  known,  the  transition-curves  may  be 
located. 

EXAMPLE. — A  6°  curve  and  a  4°  curve  are  united  by  a  tangent 
540  ft.  long;  BHfor  6°  curve  =  4.5  ft.;  CG  for  4°  curve  =  3  ft.; 
B  is  at  sta.  180,  C  at  185  +  40.  Find  F  and  F'. 

By  (201),  tan  a  =  ~  =  .0139  =  tan  0°  48'. 

ft 

B  will  be   moved   forward  '»-  =  .133  stas.  =  13.3  ft.  to  sta. 

Q 

180  +  13.3,  and  G  will  be  moved  backwards  ~  =  .2  stas.  or  20 

4 

ft.  to  185  +  20. 

By  (202),        F=  (955.4  +  4.5)0.99990  -  955.4  =  4.4  ft. 
By  (203),       F'  =  (1432.7  +  3)0.99990  -  1432.7  =  2.86  ft. 

These  values  call  for  li  =  317.8  ft.  for  6°  curve,  and  Z,  =  312.7 
for  4°  curve. 

REMARK.— It  will  frequently  be  found  that  tbis  problem 
allows  tbe  line  to  be  thrown  on  better  ground.  Should  the 
ground  require  tangent  to  be  shifted  inward,  the  curves  must 
be  sharpened  by  compounding  to  admit  of  the  necessary  offsets. 

163.  Having  Run  a  Tangent  which  Falls  Outside  a  Located 
Curve,  to  Find  the  Offset  F  for  a  Transition-curve  Uniting 
them. 

In  Fig.  76  let  the  tangent  be  AB  ;  CE  the  located  curve.     Set 
transit  at  some  point  C,  and   bring   telescope  into  tangent  to 
curve.     Measure  CB  and  move  to 
B,    where    angle    ABC  must    be    " 
measured  ;    or    measure    CH  per- 
pendicular to  AB;  then 

CH 
sm«  =  _ 

Now  EG  —  R  vers  a  ;  or  it  is 
the  rnid-ordinate  for  twice  a,  and 
may  be  found  from  Table  IX ; 
then 


=  CH-  EG  =  CH  -  It  yei-s  a.     .     .     .    (204) 
ound  from  C  by  the  relati< 
The  transition-curve  may  now  be  located. 


The  point  E  is  found  from  C  by  the  relation  EC  —  — • 


140     A   FIELD-MANUAL  FOR   RAILROAD   ENGINEERS. 

164.  Inserting  Transition-curves    in    Old    Track.  —  Sections 
159  and  160  afford  the  means  of  inserting  transition-curves,  of 
which  159  is  theoretically  the  best,  though  from  the  amount  of 
track  disturbed  it  may  be  better  to  employ  160.     Sometimes  the 
method  of  162  maybe  employed  to  advantage  when  the  connect- 
ing  tangent   is  short.     For  easing  the  curves  at  point  of  com- 
pounding, the  method  of  161  may  be  made  use  of. 

The  offsets  must  necessarily  be  small  if  the  new  track  is  re- 
quired to  occupy  the  old  road-bed.  It  may  be  profitable  to  add 
to  the  road-bed  when  sufficient  offset  cannot  be  secured  for  sharp 
curves,  though  ordinarily  much  good  can  be  accomplished  even 
when  the  new  track  is  restricted  to  the  old  road-bed. 

Unless  the  theoretical  P.O.,  P.C.C.,  and  P.  T.  have  been 
marked  by  monuments  it  may  be  difficult  to  retrace  the  old 
lines.  If  there  is  plenty  of  room,  the  terminal  tangents  may  be 
prolonged  to  intersection  and  /measured,  after  which  the  degree 
of  curve  may  be  found  by  measuring  around  curve  and  by  ap- 
proximate measurements  of  the  tangent  distances  ;  then  one  or 
two  assumptions  and  computations  will  generally  suffice. 

In  cuts  and  rough  country  the  curve  may  be  run  out  by  setting 
transit  in  center  of  road-bed  and  measuring  the  deflection  -angles 
for  a  few  points  around  the  curve. 

After  the  transition-curves  have  been  inserted  permanent  monu- 
ments should  be  placed  at  each  end  of  transition-curve  to  guide 
the  trackman  in  keeping  up  the  proper  superelevation  of  outer 
rail. 

165.  Remarks  on  Tabular  Interpolations.  —  The  general  inter- 
polation formula  given  in  algebra  is 


_ 

-8) 


4  ! 

in  which  t  is  any  term,  a  the  first  term  taken,  p  the  number  of 
terms  from  a  to  t,  dt  the  first  from  a  of  the  first  order  of  differ- 
ences, di  the  first  of  the  second  order  of  differences,  etc. 

In  ordinary  linear*  interpolation  all  terms  after  the  second  are 
neglected  ;  in  interpolating  by  second  differences  all  after  the 
third,  etc. 

In  Tabl    XIV  linear  interpolation  will  answer  for  C  and  ordi- 


TRANSITION-CURVES.  141 

narily  for  E,  though  second  differences  may  sometimes  be  needed 
for  the  latter. 

In  Table  XV,  A  is  a  quadratic  function  of  n,  as  shown  by  for- 
mula (153),  while  B  is  a  cubic  function  of  that  portion  that  has 
been  retained.  Hence  A  should  be  interpolated  by  second  differ- 
ences, while  theoretically  B  should  be  interpolated  by  third 
differences  ;  but  as  B  is  always  quite  small,  its  second  and  third 
differences  will  be  too  small  to  affect  results,  and  linear  interpola- 
tions may  be  made  when  any  are  needed. 

In  Table  XVI  linear  interpolations  will  generally  suffice, 
though  when  F  and  y  are  large  it  may  be  necessary  to  use  second 
differences. 

The  examples  of  158  and  161  illustrate  the  method  of  inter- 
polating in  Table  XVI  for  intermediate  values  of  F  and  D 
Values  of  Fvrere  first  found  for  the  given  degree  of  curve  and 
assumed  values  of  ll  ,  so  taken  that  the  true  l\  should  be  between 
them.  From  these  assumed  values  of  l\  and  F,  taken  with  the 
required  F,  the  true  ll  was  found  by  linear  interpolation. 

As  an  extreme  case  suppose  F  and  yi  wanted  for  an  18°  curve 
when  /!  =  408  feet. 

First  write  a  few  values  of  y*  and  F  so  as  to  obtain  the  first 
and  second  differences. 


dt 


400 

81.43 

20.63 

8.08 

2.08 

420 

89.51 

0.35 

22.71 

0.10 

8.43 

2.18 

440 

97.94 

0.35 

24.89 

0.10 

8.78 

2.28 

460 

106.72 

27.17 

By  the  interpolation  formula,  when  I,  =  408, 

y,  =  81.43  +  ft  X  8.08  +  A(^\~  1}  X  0.35  =  84.62, 


F=  20.63  +  -28u  X  2.08  +  *g*°0  x  0.10  =  21.45. 

/& 

By  linear  interpolation,  y}  =  84.66,  F  =  21.46. 


14'3      A    FIELD-MANUAL,   FOR    RAILROAD    ENGINEERS. 
Again,  suppose  y<i  to  be  wanted  when  h  —  430.  By  the  formula, 


/30     _    ~\\ 

,  =  81.43  -t  ffl  X  8.08  -f  2-  ^^-  J  X  0.35  =  93.68, 


or 

y,  =  89.51  +  ig-  X  8.43  +  ™**~       X  0.35  =  93.68. 


CHAPTER  V. 

FROGS  AND  SWITCHES. 

ARTICLE  14.     TURNOUTS. 
A.   Turnouts  from  Straight  Lines. 

166.  A  Turnout  is  a  track  used  in  leaving  the  main  line.     A 
Frog  is  placed  at  the  intersection  of  main  and  turnout  rails. 

a.  The  Gauge-line  is  taken  as  coinciding  with  inside  face  of 
rail.     In  making  measurements  between  tracks  the  distance  be- 
tween corresponding  gauge-Hues  is  what  is  wanted. 

b.  The  Gauge  of  track  is  the  distance  between  gauge-lines  of 
the  rails  of  that  track. 

c.  The  Point  of  Switch  is  the  point  at  which  the  turnout  curve 
begius  ;  for  a  point  switch  (split  switch)  this  is  at  the  head-block, 
while  with  a  stub  switch  it  is  the  length  of  the  switch-rail  back 
of  the  head-block,  which  is  at  the  toe  of  switch. 

d.  The  Frog-point  is  at  the  intersection  of  the  gauge-lines  of 
intersecting  rails,  and  lies  a  few  inches  in  front  of  the  blunt 
point  of  frog  as  manufactured. 

The  angle  formed  by  the  intersecting  gauge-lines  is  the  Frog- 
angle. 

e.  The  Frog-number,  JV,  is  the  ratio  of  the  axial  length  to  the 
width  of  base  of  frog. 

,c 

E 

B 

FIG.  77. 
In  Fig.  77, 


CB  ~  w 


143 


144     A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 


Letting  the  frog-angle  BA  C  be  F,  the  figure  yields 


cot  \F  =  —  =  2N. 

*  l»n 


(205) 


(206) 


/.  The  Lead,  I,  is  the  distance  from  point  of  switch  to  point  of 
frog,  measured  along  that  main  rail  in  which  the  frog  is  placed. 
In  Fig.  78,  CB  =  I. 

g.  The  Stub-lead,  s.l. ,  is  the  distance  along  main  rail  from  frog- 
point  back  to  a  point  where  the  turnout  rail  diverges  from  main 
rail  an  amount  equal  to  the  throw.  In  Fig.  78,  KB  =  s.l.  =  I'— 
length  of  switch-rail. 

h.  The  Throw,  t,  of  switch-rail  is  the  distance  the  point  of  a 
split  switch,  or  toe  of  stub  switch,  is  moved  in  opening  or  closing 
the  switch.  A  distance  of  from  5  to  5|  inches  is  needed  to  give 
necessary  clearance  for  flanges. 

k.  The  Frog-distance,  f.d.,  is  the  length  of  the  chord  of  outer 
rail  of  turnout  from  the  point  of  a  split  switch,  or  toe  of  stub 
switch,  to  the  point  of  frog. 

167.  Given  the  Frog-number,  N,  and  the  Gauge,  g,  of  a 
Turnout  from  a  Straight  Line,  to  Find  the  Lead,  I,  and  Radius, 
jR,  of  Center  Line  of  Turnout. 

A  Q     E 


In  Fig.  78,  AC  =  g,  CB  =  I,  angle  ABC  = 

From  the  figure,  I  =  g  cot  $F. 

But,  (206^  cot  \F  =  2N. 


(207) 


FROGS    AND    SWITCHES.  145 

From  triangle  OBG, 

•    (R  +  fr)9  -(R-  M2  =  ?  =  V-y*. 

whence  2gR  =  4g*N*. 

.-.     R=2gtf*  =  ltf.     ......       (208) 


er>oA 

Taking  R  =  —   -,  inserting  in  (208),  and  solving  for  D, 


For  g  =  4  ft.  8|  in.  these  formulas  become 

2    =  9.42^  feet, (207') 

R  =  9.42JV2  feet, (208') 

D  =  ~  degrees; (209') 

and  for*/  =  4  ft.  9  in., 

I  =  9.5JV, (207") 

R  =  9.5JVr2, (208") 

Ds:  jp? C209") 

If  the  frog-distance  AB  is  wanted,  we  have  AB  =  |/J*  -f  g*,  or 
f.d.  =  g  i/4J^2  +  1  =  g  cosec  \F.       .    .     .     (210) 

EXAMPLE.— Find  I,  R,  D,  and  f.d.   for  a  No.  8  frog  and  4.75 
feet  gauge. 

By  (207"),  I  =  9.5  X  8  =  76.0  ft. ; 

B"),  R=  9.5  X  64  =  608ft.; 


«(209"),  ?-%=XMt 

«  (210),  f.d.  =  4.75  j/357  =  76.14  ft. 


14G     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

168.   Given  7?  (or  1))  and  g,  to  Find  N,  I,  and  F. 
From  (208)  and  (209), 

,fll        ./5730        53.52 
From  (207)  and  (211), 


5  =  \/2gR  =  107  y  jy       .     (212) 
From  (206)  and  (211), 

F  may  also  be  found  from  triangle  OBC,  Fig.  78  : 

cos  F  ~ (v  14} 

169.  To  Find  the  Length  of  Switch-rail,  5,  when  the  Frog- 
number,  N,  the  Throw  of  Switch,  t,  and  the  Gauge,  y,  are 
Given. 

In  Fig.  78,  by  geometry, 

HG    =    - 


i<7)  +  UG' 
Neglecting  the  EG  in  denominator  as  small, 

UG  -      ~^ 

~  2(H  +17)' 

YI  JS  2 

In  like  manner,      KL  =  — — — - 

A(M  —  %g) 

Writing  AG  =  Aff=  CK  —  <S,  and  taking  the  mean  of  de- 
nominators, 

-I 


whence  S  =  y2Kt  =  22T  i/fft.      ....  (215) 

Writing  7?  =  *i-^-, 

(216) 


FROGS   AND    SWITCHES. 


170.  Given  the  Main  Frog-number,  JV,  to  Find  the  Num- 
ber, JVi  ,  and  Lead,  ^  ,  of  Crotch-frog  for  a  Turnout  from  Both 
Sides  of  Straight  Main  Track. 

In   triangle    OC1I,   Fig.    79,  re-  0' 

memberiug  that  R  =  2gN'*, 


cos  \F,  = 


2! 


,.(217) 


Then,  by  (206), 


,.  .     .     .     (218) 


From  the  figure  and  (205), 


/?  TV"2 

^ff;    (219) 


also, 


[ (220) 

Equating  these  values  of  li  and  solving  for  JV,  gives 


(221) 


If  the  \  in  denominator  be  neglected  as  small  compared  with 
becomes 


2V,  =  ~^=  0.7072V. (222) 

If  in  (220)  we  neglect  the  \  under  radical,  there  results 

X2  =  \AUgN  =  0.707^.     .     .     .     (223) 


The  distance  between  main  and  crotch  frogs  measured  along 
main  rail  is 


-  g  12W*  +  ±,    .     .     .     .     (224) 
or,  approximately, 

I  -  I,  =  2gN  -  \AUgN  =  Q.SSGgN  =  0.293^.         (225) 


148     A    FIELD-MANUAL    FOR   RAILROAD   ENGINEERS. 


171.  To  Find  the  Radius,  R,  of  Turnout  and  Lead,  2,,  of 
Crotch-frog  in  Terms  of  the  Crotch-frog  Number,  NI 

From  (222),  .2V2  =  2JVia. 

Insert  this  in  (208)  and  (219),  giving 


(226) 
(227) 


REMARK.—  In  general  the  frogs  kept  in  stock  by  manufacturers 
do  not  afford  suitable  combinations  of  numbers  for  double  turn- 
outs. For  instance,  the  theoretical  number  of  crotch-  frog  for  a 
number  8  main  frog  is,  by  (221)  or  (222),  Ni  =  5.66,  and  we  should 
be  compelled  to  use  a  number  5£  or  6  for  the  crotch-frog;  this 
would  necessitate  a  different  rate  of  curvature  fvom  crotch  to 
main  frog  than  from  head-block  to  crotch. 

172.  Given  the  Numbers  of  Middle  Frog,  NI  ,  and  of  Main 
Frogs,  ^Vand  N',  to  Find  the  Radii  #,  from  Point  of  Switch 
to  Crotch-frog,  and  R  and  Rf,  from  Crotch  to  Main  Frogs. 

In  Fig.  80  we  have,  by  (226), 

OlN=Rl  = 
and,  by  (227), 

NC=ll= 

Now  if  Fi  ,  F,  and  F'  are  the  an- 
gles of  the  frogs  Ni  ,  Nt  and  N'  , 
the  angle 

COH=  F-  $Flt 
and 


Since  CO  —  \g,  the  triangle  CEO 
yields 

GH  =  \g  col  \(F  +  W).   .    (228) 
But,  by  trigonometry, 


, 

cot 


FROGS   AND    SWITCHES.  149 

Assume  tan  ±Fi  =  |  tan  ^Fl  ,  and  write 

taniF  =  A,    itanifl^.J^-^. 
and  after  simplifying  and  reducing, 


The  last  term  is  quite  small,  rarely  amounting  to  as  much  as 
one  inch,  and  may  be  neglected  ;  then 

--2gNNt  W  IK 

~  ~       ~         -  '  * 


From  the  triangles  LCO  and  KIIO, 

(B  +  \ff)  cos  iFt  -  (R  -f  !</)  cos  F  =  \g, 
whence 


In  like  manner  for  the  curve  CE, 
ME  =  \g  cot  \(F' 


-  x-      •         • 

2(cos  4.Fi  —  cos  F) 

EXAMPLE.—  Given  Ni  =  6,  N=  S,  and  ^'  =  9,  to  find  the  lead 
li,  the  distances  GH  and  ME,  and  radii  /?,  T^j  ,  and  K'  ,  g  being 
4.75  ft. 

By  (226),    Ri  =  19  X  62  =  684  ft.,  an  8°  23'  curve. 
By  (227),      I,  =9.5  X  6  =  57ft. 

By  (230),  GH  =  —  jfq^  -  22.8  ft. 
By  (232),  ME  =  24.4  feet. 


150     A    FIELD-MANUAL    POK    RAILROAD    ENGINEERS. 


4.75 


By  (231),     *=2(cos4o46,_cosr9<} 


-— _  -  2.38  =  547.4  ft., 


a  10°  28'  curve. 

By  (233),.  R'  = 
a  6°  32£'  curve. 


4.75 


2(cos  4°  46'  —  cos  6°  22  ) 


-  2.38  =  876.4  ft. 


173.  Given  the  Number,  N,  of  the  Two  Main  Frogs  and  the 
Gauge,  g,  to  find  the  Crotch-frog  Number,  JV, ,  its  Lead,  I,  ,  and 
the  Radius,  J^i  ,  of  Curve  through  Crotch  when  the  Double 
Turnout  is  to  Same  Side  of  Straight  Main  Track. 

In  Fig.  81  the  frogs  at  B  and  O  are  of  the  same  number,  and 
may  be  taken  as  falling  on  the  same  straight  line  through  the 
center  0.  Angle  0,£0  =  90°  —  OGL  =  F,  and  the  triangle 
00 iG  is  therefore  isosceles;  hence 


0,0  =  0,0  =  OA  -  0,0  =  IOA, 


or 
whence 


(234) 


Now,  by  the  same  reasoning  as  in  167, 


=  Rlt  whence 

m 


FROGS   AND    SWITCHES. 


151 


Neglecting  the  |  under  radical  and  writing  R  =  2gN*  gives 

&  =  ^-_  =  .707JV,    ....  (236) 

f/2 

which  is  identical  with  (222)  for  turnouts  to  opposite  sides.     For 
EG  and  EB,  as  in  167,  I,  —  2gNi  and  I  =  2gN.     Hence 


CB  =  I  -  I,  =  2g(N~  -  NJ  .....     (237) 

EXAMPLE.  —  Find  Ni,    Ri,  and   I  —  li   where    .ZV  =  9    and 

g  =  4.75  ft. 

By  (208"),     R    =  9.5  X  81  =  769.5  ft.,  a  7°  27'  curve. 
"  (234),      Ri   =  384.75  -  1.19  =  383.56  ft.,  a  14°  56'  curve. 
"   (236),      Ni  =  .707  X  9  =  6.36. 
"  (237),       CB  =  I  -  I,  =  9.5  X  2.64  =  25.08  ft. 

REMARK.  —  It  may  now  be  seen  that  the  proper  combination  of 
frogs  for  a  double  turnout  to  opposite  sides  applies  also  where 
the  turnouts  are  to  same  side  of  straight  main  line.  Also  they 
apply  to  turnouts  from  opposite  sides  of  curved  main  line  when  its 
radius  is  not  less  than  that  required  by  main  frog  for  straight 
track. 

174.  Given  the  Number  of  Main  Frogs,  N,  and  of  Crotch- 
frog,  N\  ,  to  Find  the  Radius  of  Curve  between  Frog-points  of 
a  Double  Turnout  to  Same  Side  of  Straight  Track. 


IE 


0, 


0 

FIG.  82. 

In  Fig.  82,  02  £  =  R*  -f-  \g,  and  the  chord  CO  must  be  deter- 
mined. The  frogs  at  B  and  G  being  of  the  same  number, 
0a (70  =  GOO,  =  F  and  CO,E  =  F,. 


152     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


Draw  GH  perpendicular  to  EB\  then  in  triangle  BGH 
GH  =  g  cos  F. 

Draw  OiL  perpendicular  and  GK  parallel  to  EB  ;   from  tri- 
angles 02  O  K  and  0*CL, 


-f  i#)(cos  Ft  -  cos  2F)  =  KL  =  OH  =  g  cos 


whence 


+  fc  = 


#  cos  F 


cos  .F,  -  cos  2F  ' 
From  triangle  0ZCG,  since  <702£  =  2^—  Flf 

CO  =  2(11*  +  \g)  sin  |(2F  -  F}).  . 


(238) 


.      (239) 

EXAMPLE.—  Given  .ZV  =  8,  2Ti  =6,  and  </  =  4.75,  to  locate  the 
turnout. 

By  (208),  R  =  608  ft.  ;  R1  =  342  ft. 

By  (238),  S,  +  \g  =  274.5  ft. 

By  (239),  GO  =  22.8  ft. 

175.  Given  the  Frog-number,  N,  the  Gauge,  g,  and  Distance, 
p,  between  Centers,  to  Unite  Main  Line  with  a  Parallel  Siding 
when  the  Reversing-point  is  at  Frog-point. 

0|V 


In  Fig.  83,  BOi  =  Ri  —  ^g  and  BE  are  required. 
In   triangle  BO.E,  BO,  -  Rv  -  \g,  E0>  =-  R,  -f 
angle  BO^E  =  JP1.     By  trigonometry, 


p,  and 


PROGS   AND   SWITCHES.  153 

(Ri  -  iff}  +  (Si  +  $ff-P)  =  tan  ^(180°  -  F) 
(Si  -  iff)  -  (Ri  +  ]0-p)~          tan  IF 


+  \p,      .....    (240) 


whence 


T>Tf  7 

From  the  similar  triangles  ABG  and  BCE,  -  =  -  ,  from 

P-ff       9 

which 

(242) 


EXAMPLE.—  Find  Rl  and  BE  when  N  =  8,  p  =  12.35  ft., 
g  =  4.75  ft. 

By  (240),      R,  =  15.2  X  64  +  6.2  =  979  ft.,  a  5°  51'  curve. 

By  (207"),        I  =  9.5  X  8  =  76  ft. 

By  (242),     BE  -  (2.6  -  1)  X  76  -  121.6  ft. 

REMARK.  —  If  space  requires  that  the  turnout  get  away  from 
main  line  more  rapidly  than  by  the  above  method,  we  can  assume 
the  second  radius  equal  to  or  less  than  the  radius  of  turnout  and 
find  the  re  versing-  point  by  131,  and  then  compute  BE. 

176.  To  Lay  Out  a  "  Ladder"  Track 

In  yardwork  a  number  of  parallel  sidiugs  may  be  conveniently 
connected  with  the  main  line  by  means  of  a  ladder-track. 

In  Fig.  84,  if  the  frog-number  .ZV  and  the  distance  2?  between 
center  lines  of  track  are  given,  it  is  only  necessary  to  determine 
the  distances  BC,  CE,  etc.,  between  frog-points,  and  BK,  CL, 
etc.,  between  point  of  switch  and  point  of  frog.  From  triangle 
BCG, 


,      .......     (243) 

BK  =  BC  -  KG  =  p  cosec  F  -  2gN  ;  .     .     .      (244) 


154     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


or,  since  cosec  F  —  N  -\-  — r,  (see  186,) 


(243') 


~ 


(244') 


EXAMPLE.—  For  a  No.  8  frog  find  BC&ud  BKwheu  p  =  12.8  ft. 
d     =  4.75  ft. 


By  (243'), 


BC  =  12.8  X  8  +  -        =  102.8  ft. 


By  (244'),          BK  =    3.3  X  8  +  0.4  =  26.8  ft. 


B.    Turnouts  from  Curves. 

177.  Given  the  Radius  of  Main  Curve,  the  Frog-number, 
and  the  Gauge,  to  Find  the  Radius  and  Lead  of  Turnout  from 
Concave  Side  of  Main  Line. 

In  Fig.  85,  AB  is  the  outer  rail  of  turnout,  CB  the  inner  rail  of 
main  track.  In  triangle  OAB,  since  0ZBA  =  GAB, 

OBA  -  OAB  =  F, 
OBA  -f  OAB  =  180°-  0, 
and     OA  =  11  +  iflr,   OB  =  R  -  Ig. 


FROGS   AND   SWITCHES. 

n  \ 

Then,  by  trigonometry, 


(R  +  iff)  +  (R  -  iff)  -  lanj(180j-_0)     j  cot 
(-R  +  W  -  (#  -  iff)  ~  ,      tan          " 


155 


Q  7?  7? 

Reducing,       cot  |6  =  —  tan  \F  =  — . 


Then 


I  =  EG  = 


....     (245) 
sinffl.       .     .     .     (246) 


If  the  length  of  AB  is  wanted,  we  can  show  that  the  angle 
ABC=±F;  and   by  solving   the  triangle  ABC,  since   ACB  = 

90°  -f  W, 


To  find  7?2  ,  from  triangle  0*AB, 

2(.R,  +  i£)smi(F4-6)  =  AB  .....     (248) 
Or,  in  triangle  BO^C, 


(K*  +  Iff)  +  (R*  -  Iff)  :    tnn|[180-(F+fl)]  _  cot  ^(F+  6) 
C^"+  ^)  -  (It*  -\g)  ^  tan  "i  A'  ~lauP      ' 


156     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 
Reducing  and  solving  for  R*t 

*  =  I  •  **££*  =  I  •  cot  *'•  cot  ^F  +  '>•  •  <349> 

But,  from  trigonometry 


Substitute  this  in  (249)  and  write 


cot  \F  =  2N,    tan  ±F  =  ^^,     tan  \B  —  '—£-, 

&jy  M 

and  reduce ;  then 


For  2gN*  write  .R, ,  the  radius  of  turnout  from  straight  track, 
and  neglect  the  \g  in  numerator  as  small  compared  with  R ;  then 


(251) 


Now  write 


D       5730      „          5730       „     _  5730 

"IT'   J       :^T'    ^-"^T 

and  reduce,  yielding 

Z>2  =  D  +  Z>!  ........     (252) 

Formula  (252)  affords  an  easy  method  of  finding  the  degree  of 
turnout  curve,  or,  if  preferred,  the  radius  may  be  first  found  by 
(251). 

Draw  OE  to  the  mid-point  of  CB  ;  0#does  not  differ  greatly 
from  OB  or  OC  ;  so,  if  we  write  OE  =  R  —  \g,  there  results 


1  =  2(11-  ft)  tan  &  =  2(R  -  \g-  =  2gN  -  ~.     (253) 

A  M 

The  last  term  is  quite  small,    even  in  the  most  extreme  case 
likely  to  arise  in  practice  ;  for  a  turnout  from  a  6°  curve  with 


FROGS    AND    SWITCHES. 


157 


number  8  frog  it  amounts  to  only  2|  inches  ;  neglecting  it.  we 
may  write,  as  for  straight  main  track, 


(254) 


EXAMPLE. — Turnout  from  inside  of  a  4°  curve,  N=S,g  = 
4.75ft. 

By  (208"),  $!  =  9.5  X  64  =  608  ft.,  a  9°  26'  curve. 

By  (252),    Z>2  =  4°  +  9°  26'  =  13°  26',  for  which  R2  =  426.8. 

By  (254),       I  =  76  ft. 

178.  Given  the  Frog-number,  the  Gauge  and  Radius  of  Main 
Curve,  to  Find  the  Lead  and  Radius  (or  Degree)  of  Turnout 
from  Convex  Side  of  Main  Line. 

In  triangle  AOB  of  Fig.  86,  A  +  B  =  180°  -  0, 

K°2 


A  -  B  =  (180°  -  09AS) 
-(18W-0*BA- 

By  trigonometry, 


_  tan  |(180  -6) 


or 
whence 


-(tf-te) 

2_R      cot 
9 


tan 


O  7? 

cot    O  =        tan    ^= 


From  triangle  OCB, 
l  =  CB  =  2(R  +  %g 
Assuming  OE  =/? 


.   (256) 


—-.  (257) 

Neglecting  the  last  term  as  small,  as  in  177, 
I  =  2gN,  . 


which  is  the  same  as  (207). 
In  triangle  CO^D,  0*  —  F  -  B. 


(258) 


.'.  tan 


158     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

We  may  now  follow  the  same  line  of  reasoning  by  which  (251) 
was  derived,  or  more  simply  by  assuming  the  tangent  of  the 
difference  of  two  small  angles  equal,  to  the  difference  of  their 
tangents;  that  is,  tan  |02  =  tan  \F  —  tan  W. 

Now  it  can  be  easily  shown  that  tan  402  =  ~  ;  therefore 


gN= 

1      gN 

2N       R  ' 

whence 

l 

1111 
2gW      R  ~  R!       R' 

from  which 

R,  - 

RR, 

R  -  R,  '     '     '     ' 

(259) 

5730    7          5730    _        5730 
Write  Ri  =  -—  ,  Rl  =  -=—,  R  =  -^=-  ,  and  solve  for  Z>3. 

X/2  JJ\  JJ 

D9  =  D,  -  2),  .......     (260) 

in  which  Di  is  the  degree  of  turnout  from  straight  track. 

EXAMPLE.  —  Turnout    from    outside    of    a    4°   curve,   N  =  8, 
g  =  4.75. 

By  (208"),   ^i  =  9.5  X  64  =  608  ft.,  a  9°  26'  curve. 

By  (260),      D2  =  9°  26'  -  4°  =  5°  26',  for  which  R  =  1054.9. 

By  (258),         I  =  9.5  X  8  =  76  ft. 

From  (255)  we  have,  by  inverting, 

OQ 

° 


ana,  by  (256), 

1  =  2870  X  sinl°  31'  =  75.97ft., 

a  difference  of  only  0.03  ft.  from  the  value  given  by  (258). 

179.  To  Find  Theoretical  Length  of  Switch-rail  when  the 
Turnout  is  from  a  Curved  Track. 

A  common  tangent  being  drawn  at  the  switch-point,  we  shall 
have,  as  in  169,  for  offset  from  tangent  to  main  curve, 


FROGS   AND    SWITCHES.  159 

the  offset  from  tangent  to  turnout  is 

_   £2 
2/2  -  2lT2- 

When  the  turnout  is  from  concave  side  of  main  line, 

y*  -  y  =  t\ 

therefore  t  =  -  I), 


whence  8 


=  j/  2tKE*  ..........     (261) 

'      R  —  jfta 


Writing  R  =  55?,  R,  =  5™   and  reducing, 

n  Hi 


When  the  turnout  is  from  convex  aide  of  main  line, 


.........     (263) 

i     --          3 

from  which 


In  (262)  and  (264)  Z>!  is  the  degree  of  turnout  from  straight 
track,  and,  as  these  formulas  are  identical  with  (216),  it  is  seen  that 
the  theoretical  length  of  switch-rail  on  turnouts  from  curves  is 
the  same  as  on  turnouts  from  straight  line. 

EXAMPLE.—  Find  8  when  t  =  0.42,  N  =  8,  g  =  4.75. 

By  (208"),     Rl  =  608  feet,  for  which  D,  =  9°  26'. 


160     A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 
By  (216),  (262),  or  (264), 

8=  107  |/—  =22.6  feet. 

180.  Given  the  Distance  p  between  Center  Lines  of  Curved 
Main  Line  and  Side  Track,  the  Frog-angle,  F  (or  Number,  N ), 
and  Gauge,  g,  to  Find  the  Radius  and  Central  Angle  of  Curve 
beyond  Frog- point. 

FIRST  CASE. — Turnout  from  outside  of  main  line. 

In  Fig.  87,  0  is  the  center  of  main  curve,  Oi  the  center  of 


curve  whose  radius  is  required.     In  triangle  BOG, 

CO  =  R  +  p  -  \g,    BO  =  R  +  \g. 
By  the  same  reasoning  as  in  177, 


:.  .     .     (265; 


-    p-g~  ZN(p-g)' 

In  triangle  00 \B,  OiB  =  Ri  —  \g\  then,  by  the  law  of  sines, 


Also, 


FROGS   AND    SWITCHES. 
SECOND  CASE. — Turnout  from  inside  of  main  track. 


161 


In  Fig.  88,  we  have  from  triangle  BOE,  reasoning  as  in  178, 


cot  40  = 


~  P 


—  g 


;  .          ,     (268) 


and  from  triangle 


Also,         £<7  =  2(7?  -  i«7)  siu  H 
and  BE  =  2(7?,  -  ^)  sin 


(269) 


(270) 
-  0)  .....     (271) 


When  0  is  greater  than  F,  siu  (F  —  0)  is  negative,  and  center 
i  falls  on  same  side  as  0,  and 


162     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


sin  (0  -  F)v" 
BE  =  2(R>  -f  4#)  sin 


-  F).  .     . 


.     (272) 
,    (273) 


C.   The  Stub  Lead. 

181.  When   the   frog-number  exceeds  seven,    the    length   of 
switch-rail  required  to  give  the  necessary  clearance  at  heel  be- 
comes greater  than  is  allowed  in  practice.     To   overcome  this 
difficulty  slightly  more  curvature  is  given  the  switch-rail  ;  more- 
over the  physical  point  of  switch  is  necessarily  some  distance  in 
advance  of  the  theoretical  point.     The   distance   from   heel   of 
switch  to  point  of  main  frog  will  then  be  the  same  as  from  head- 
block  of  stub  switch  to  main-frog  point,  and  is  termed  the  Stub 
Lead.     If  to  this  distance  the  length  of  switch-rail  be  added,  we 
get  the  distance  from  the  head-block  of  a  point  switch  to  the 
point  of  main  frog,  which  is  the  Short  Lead  required  in  practice. 

182.  Given  the  Throw,  t,  the  Gauge,  g,  and  the  Frog-number, 
N,  to  Find  the  Stub  Lead,  s.l. 

In  Fig.  89,  KB  is  the  stub  lead  required;  GN=  KL,  the  throw. 


From  (207), 
and  from  (215), 
From  the  figure, 


or 


FIG.  89. 

I  =  CB  =  2gN, 
8=  CK  =  2N  tfgt. 

KB  =  CB  -  CK, 

s.i.  =  2y((/  - 


(274) 


FUOGS   AND    SWITCHES. 


163 


Formula  (274)  may  be  employed  for  turnouts  from  curves  as  well 
as  straight  lines,  since  it  was  shown  that  the  formulas  from  which 
it  was  derived  may  be  employed  even  when  the  curvature  of  main 
track  is  considerable. 

Below  is  a  table  of  values  of  (g  —  \/gt)  for  some  of  the  more 
common  values  of  g  and  t. 

TABLE  OF  VALUES  OF  g  —  ^gt. 


3  Feet  Gauge. 

4  Feet  8i  Inch  Gauge. 

4  Feet  9  Inch  Gauge. 

Throw. 

g  -  Vgt. 

Throw. 

g  -  Vgt. 

Throw. 

g  -  Vgt. 

Inches. 
3 
8* 

Feet. 
2.13 
2.06 
2.00 

Inches. 
5 

§ 

Feet. 
3.308 
8.339 

3.206 

Inches. 
5 

H 

5J 

Feet. 
3.343 
3.275 
3.242 

EXAMPLE.—  Find  the  stub  lead  for  N=  8,  g  =  4.75  ft.,  t  =  5 
inches. 


From  the  table,       g  —  ^gi  =  3.343  ft., 
and,  by  (274),  s.L  =  16  X  3.343  =  53.49  ft 


183.  The  Turnout  Table  on  the  next  page  gives  the  frog-angles, 
the  radius  of  center  line  of  turnout  from  a  straight  track  and  its 
degree,  the  theoretical  lead,  the  theoretical  length  of  switch-rail 
for  t  =  5  inches,  and  the  stub  lead  for  certain  values  of  t.  The 
frog-numbers  given  cover  all  the  usual  cases. 

Suppose  it  required  to  find  the  short  lead  for  a  No.  9  frog  and 
5-iuch  throw  when  the  gauge  is  4  ft.  9  inches  and  the  length  of 
switch-rail  18  feet.  From  the  table  the  stub  lead  is  60.17  feet; 
hence  the  short  lead  is  60.17  +  18  =  78.17  feet,  as  against  85.50 
ft.  for  the  theoretical  lead. 

Inspection  of  the  table  will  show  that  it  makes  no  very  great 
difference  in  the  tabular  quantities  whether  the  gauge  be  taken 
-is  4  feet  8£  inches  or  4  feet  9  inches.  However,  the  numerical 
coefficients  in  the  formulas  involving  g  are  somewhat  simpler  for 
the  latter  value, 


164     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


TURNOUT  TABLE  FOR  STRAIGHT  TRACK. 

4  FEET  Sya  INCH  GAUGE. 


Frog 

No. 

Frog 
Angle. 

Theo- 
retical 
Lead. 

Turn- 
out 
Radius. 

Degree 
of 
Turn- 

Theoret- 
ical 
Switch- 
rail  for 

Stub-lead  for  a  Throw 
of 

out. 

i  =  5In. 

5  In. 

5^  In. 

5%  In. 

0               / 

feet 

feet 

0               / 

feet 

feet 

feet 

feet 

4 

14    15 

37.67 

150.7 

38    2 

11.20 

26.46 

25.91 

25.65 

5 

11    25 

47.08 

235.4 

24  21 

14.01 

33.08 

32.39 

32.06 

^ 

10    28 

51.79 

284.9 

20    7 

15.41 

36.39 

35.63 

85.27 

6 

9    32 

56.50 

339.0 

16  54 

16.81 

39.70 

'38.87 

38.47 

W 

8    48 

61.21 

397.9 

14  24 

18.21 

43.00 

42.11 

41.68 

rf 

8    10 

65.92 

461.4 

12  25 

19.61 

46.31 

45.35 

44.88 

r^ 

7    38 

70.63 

529.7 

10  49 

21.01 

49.62 

48.59 

48.09 

8 

7      9 

75.33 

602.7 

9  30^ 

22.41 

52.93 

51.82 

51.30 

8^ 

6    44 

80.04 

680.4 

8  25 

23.81 

56.24 

55.06 

54.50 

9 

6    2-1 

84.75 

762.7 

7  31 

25.21 

59.54 

58.30 

57.71 

9^ 

6      2 

89.46 

849.8 

6  45 

26.61 

62.85 

61.54 

60.90 

10 

5    44 

94.17 

941.7 

6    5 

28.01 

66.16 

64.78 

64.12 

11 

5    12 

103.58 

1139.4 

5    2 

30.81 

72.78 

71.26 

70.53 

12 

4    46 

113100 

1356.0 

4  13J^ 

33.61 

79.39 

77.74 

76.94 

13 

4    24 

122.42 

1591.4 

3  36 

36.42 

86.01 

84.21 

83.36 

14 

4      5 

131.83 

1845.7 

3    6 

39.22 

92.62 

90.69 

89.77 

15 

3    49 

141.25 

2118.7 

2  42 

42.02 

99.24 

97.17 

96.18 

4  FEET  9  INCH  GAUGE. 


Frog 
No. 

Frog 
Angle. 

Theo- 
retical 
Lead. 

Turn- 
out 
Radius. 

Degree 
of 
Turn- 

Theoret- 
ical 
Switch- 
rail  for 

Stub-lead  for  a  Throw 
of 

t  =  5  In. 

5  In. 

5J^j  In. 

b%  In. 

0             / 

feet 

feet' 

0          / 

feet 

feet 

feet 

feet 

4 

14    15 

38.00 

152.0 

37  42 

11.26 

26.74 

26.20 

25.94 

5 

11     25 

47.50 

237.5 

24    8 

14.07 

33.43 

32.75 

3-,>.42 

10    23 

52.25 

287.4 

19  56 

15.48 

36.77 

36.03 

35.66 

6  ^ 

9    32 

57.00 

342  0 

16  46 

16.88 

40.12 

39.30 

38  90 

8    48 

61.75 

401.4 

14  16 

18.29 

43.46 

42.58 

42.15 

7 

8    10 

66.50 

465.5 

12  19 

19.70 

46.80 

45.85 

45.39 

71^ 

7    38 

71.25 

534.4 

10  44 

21.10 

50.15 

49.13 

48.63 

8 

7      9 

76.00 

608.0 

9  25^ 

22.51 

53.49 

52.40 

51.87 

6    44 

80  75 

686.4 

821 

23.92 

56.83 

55.68 

55.11 

9 

6    22 

85.50 

769.5 

7  27 

25.32 

60.17 

58.95 

58  36 

6      2 

90.25 

857.4 

6  41 

26.73 

63.52 

62.23 

61.60 

10 

5    44 

95.00 

950.0 

6    2 

28.14 

66.86 

65.50 

64.84 

11 

5    12 

104.50 

1149.5 

4  59 

30.95 

78.55 

72.05 

71.32 

12 

4    46 

114.00 

1368.0 

4  11 

33.77 

80.23 

78.60 

77.81 

13 

4    24 

123.50 

1605.5 

3  34 

36.58 

86.92 

85.15 

84  29 

14 

4      5 

133.00 

1862.0 

3    4}/£ 

39.39 

93.60 

91.70 

90.78 

15 

3    49 

142.50 

2137.5 

2  41 

42.21 

100.29 

98.25 

97.26 

I 

FROGS   AND   SWITCHES.  165 

184.  To  Stake  Out  a  Turnout.  —  If  the  position  of  head-block 
is  given,  fix  the  frog-point  by  the  foregoing  table,  remembering 
that  it  may  be  used  for  turnouts  from  curves,  as  well  as  from 
straight  lines,  without  material  error. 

To  locate  the  rail  between  head-block  and  point  of  switch  it  is 
sufficient  to  do  so  by  offsets  from  main  rail.  Consider  the  equa- 
tion (36)  for  tangent  offsets. 

z  =  lri*D  for  straight  main  line. 

2  =  |/i'-'(Z>i  ±  D)  for  curved  main  line. 

At  frog-point  z  =  g,  and  n  =  n.  ;  hence 

g  =  liifD,   or    ln,\D,  ±  D). 


At  mid-point  of  curve  (practically  mid-point  of  lead),  n  =  £»,  , 
and 

z  =  l.  $nSD,  or  £  .  ^(D,  ±  D)  =  \g.  .  (275) 
When  n  =  \nl  , 

z  =  |  .  ^n^D,  or  £  .  ^nS(D>  ±  D)  =  ^g.  .  (276) 
When  n  —  \nlt 

z  =  |  .  &nSD,     or    |  .  &»»(£,  ±  D}  =  fig.    .     (277) 

These  formulas  are  for  the  theoretical  lead,  and  afford  an  easy 
method  of  locating  the  outer  rail  of  turnout  with  all  the  accuracy 
needed  in  practice. 

185.  Curving  Rails.—  In  bending  rails  for  curves  the  proper 
curvature  is  determined  by  measuring  the  mid-ordinate  from  a 
cord  held  against  the  inside  face  of  rail-head. 

This  ordinate  may  be  determined  by  (18),  in  which  n  is  the 
half-length  of  rail  divided  by  100.  For  a  30-ft.  rail, 

M  =  |(0.15)2J9  =  0.0196Z>  =  0.02D  (nearly).     .     (278) 

From  (209'),  D  =  ~;  and  from  (209"),  D  =^-  Inserting 
either  of  these  values  in  (278)  gives 

M=~,  "early  .......     (279) 


166     A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

When  the  turnout  is  from  a  curve  compute  M  from  (279),  and 
the  mid-orclinate  for  a  rail  30  ft.  long  on  main  curve  by  (278); 
then  the  mid  ordinate  for  turnout  rail  will  be  the  sum  or  differ- 
ence of  these  values  according  as  the  turnout  is  from  concave  or 
convex  side  of  main  curve 


ARTICLE  15.  CROSSOVERS. 

186.  To  Locate  a  Crossover  between  Parallel  Straight 
Tracks  when  the  Frog-number,  the  Distance,  p,  between  Cen- 
ters, and  the  Gauge  are  given,  inserting  a  Tangent  between 
Frog-points. 


FIG.  90. 


In  Fig.  90  it  is  required  to  find  OB  =  KG  =  I,  MK  and  NC, 
sdso  HK  =  k. 

In  the  triangle  BPM,  BM=  p  —  g;  then 


or 

and 

or 


BE-k-BP-  EP 

k  —  ( p  —  g)  cosec  F  —  g  cot  F, 
MK=MP-  KP, 
MK  =  (p  —  g}  cot  F  —  g  cosec  F. 


(280) 


(281) 


From  triangle  OBC  of  Fig.  78, 


_OC_  R-  l-g  _2R-  g 

~  OB  ~  2T       ~  2F+7 


FROGS   AND   SWITCHES.  167 

In  («)  write  U  =  ZgW  by  (208),  giving 


-  g      4*-I 

=  ..... 


From  Fig  78,  triangle  OBC, 

GB  t  21 


'     ' 


Writing  I  =  2gN  and  E  =  2gN*  gives 


From  trigonometry,  taking  the  above  values  of  sin  ^7aud  cos  F, 


Inserting  these  values  in  (280)  and  (281), 

(282) 

(283) 
By  (207),  GB  =  KG  =  I  =  2glf;  therefore 

NC=  21  +  MK=4gN+  (p  -  2g)N  -       , 

N-.  .         (284) 


EXAMPLE.—  Find  k  and  MK  for  a  No.  8  frog  when  p  —  13  ft. 
and#  =  4.75  ft. 

By  (282),  k  =  3.5  X  8  -p  0.4  =  28.4  feet. 

By  (283),       MK  =  3.5  x  8  -  0.4  =  27.6  feet. 


168      A    FI  ELD-MA  NTAL    FOR    KA1  I/ROAD    KNGINKERS. 

187.  To  Lay  Out  a  Crossover  in  the  Form  of  a  Reversed 
Curve. 

When  p  is  large,  or  for  other  reasons  it  is  desirable  to  get  away 
from  mam  track  more  rapidly  than  by  the  foregoing  method,  we 
may  lay  out  the  crossover  in  the  form  of  a  reversed  curve. 


FIG.  91. 


In  Fig.  91  it  is  required  to  find  GB  —  HE  and  LH. 
Find   OB  =  HE  =  I  by  (207),  and  the  radius  OC  = 
(208). 

Then,  from  (113),  we  have 

ME  =  2R  sin  a. 
The  angle  a  is  given  by  (112).     Then 

LH  is  2R  sin  a  -  21. 


by 


(285) 


188.  To  Lay  Out  a  Crossover  when  a  Fixed  Length  of  Tan- 
gent   must  be    Interposed   between    Points    of    Reversal    of 
Curvature. 

From  the  given  frog-number  determine   the  radius  by  (208)- 
then  the  problem  may  be  solved  by  132. 

189.  To  Lay  Out  a  Crossover  in  the  Form  of  a  Reversed 
Curve  when  the  Tracks  to  be  Joined  are  Curved. 

In    Fi       92    let    the   notation    be   as   shown.      Let    OM  =  11, 


0,  02  =  7?,  +  7?2  =  a, 
00?    =  E  +  p  -  R,  =  b, 


FROGS   AND   SWITCHES. 


169 


00,    =  R  +  Ri  =  c. 
i(a  +  5  +  c)  =  «. 


0 

FIG.  92. 
Then,  from  trigonometry, 

cos  \A  — 


>-q) 
be 


8(8  -  b) 


(286) 


(287) 


Iti  is  determined  by  178,  and  7?2  by  177,  while  R  and  p  are 
given  to  begin  with. 

The  angle  (A  +  B)  determines  the  length  of  arc  CP,  and  angle 
B  the  length  of  arc  HP. 

The  angle  6  is  given  by  (255),  and  Bi  by  (245).  Hence  angle 
G01I,  which  determines  the  arc  (7//measured  along  main  track 
between  frog-points,  is 

GOII=  A  -  (0  +  6,). 

The  frog-numbers  at  G  and  B  need  not  be  equal,  only  providing 
that  P  falls  between  G  and  B. 


170     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


ARTICLE  16.     CROSSING-FROGS  AND  CROSSING  SLIPS 
A.    Crossing-frogs. 

190.  When  two  tracks  intersect  each  other  four  crossing-frogs 
are  required  at  the  intersection  of  the  two  sets  of  rails.    The  four 
frogs  are  sometimes  called  a  set  of  crossing-frogs. 

191.  To  Find  the  Length  of  Rails  Intercepted  between  two 

Intersecting  Straight  Tracks  when 
the  Angle  of  Intersection  and  the 
Two  Gauges  are  given. 
In  Fig.  93,  from  triangle  ABH, 


AB  =  EC  =  g  cosec  F ;  .     (288) 
and  from  triangle  AEG, 

AE  =  BC  -  gt  cosec  F.       (289) 


FIG.  93. 


192.  Given  the  Angle  of  Intersection,  a,  made  by  the  Center 
Lines  of  a  Straight  and  Curved  Track,  the  Gauges  gi  and  g,  to 
Find  the  Angles  of  the  Set  of  Crossing-frogs. 

In  Fig.  94,  from  the  triangles  OBK  &ud  OAH, 

(R  +  i<?)  cosF=R cos  a  -f  ^,. 
R  cos  a  -f-  iff, 


.-.  cos-F^ 

In  like  manner, 

R  cos  a  — 


(290) 


cos  F*  = 


cos  Z3  = 


R  cos  a  — 
R-±o 


N  M   HK      L. 

FIG.  94. 


From  triangle  BOG  to  find  the  chord  BC. 

BC  =  2(1?  -f  |<?)  sin  '(#  -  F). 


(294) 


FROGS   AND   SWITCHES. 


171 


Similarly, 

GE  =  2(R  -  \g]  sin  l(F*  -  F,) (295) 

From  triangles  EOM  and  COL,  we  have 

EC  =  ML  =  (R  +  y)  sin  F,  -  (R  -  $g)  sin  F2.  (296) 
In  like  manner, 

GE  =  NK  =  (R  +  i<7)  sin  F  -  (R  -  %g)  sin  F3.      (297) 

193.  Given  the  Angle  of  Intersection,  a,  made  by  the  Center 
Lines  of  Two  Curved  Tracks, 
their  Gauges,  g  and  gt  ,  to  Find 
the  Angles  of  the  Crossing-frogs. 

In  Fig.  95,  OA  =  R,  0,A  =  ft, , 
and  angle  0J.01  =  a  of  the  triangle 
OAOi  are  given;  whence  00i  may 
be  determined. 

In  triangle  OBOl  the  side  OB  = 
R  +  lg,    0,£  =   1?,   -f  \9l,    and 
00!   =  k  are  known,   from  which 
we  can  determine  the  angle  0^?0,  = 
FIG.  95.  F. 

In  like  manner  from  the  triangle    OCOl  determine  Fit  and 
from  triangle  0^0!  find  F9.  F3  may  be  found  from  triangle  OGOi. 

To  find  the  chord  GB  first  find  angle         Q 
S0j  0    from  triangle  B0l  0,  and  angle 
GOiO  from  triangle  GOiO;  then 


GB  =  2(R,  -f  £0,)  sin 
In  like  manner, 


(298) 


EC  =  2(1?!  -  ^,)  sin  $#0!  (7,  (299) 
BC  =  2(R  -f  £0)  sin  *BOC,  .  (300) 
^^  =  2(R  -  \g}  sin  *GOE.  .  (301) 

When  the  tracks  intersect,  as  in  Fig. 
96,  the  solution  is  evidently  similar  to 
the  foregoing, 


FIG.  96. 


A   FIELD  MANUAL   FOR   RAILROAD   ENGINEERS. 


B.   Crossing-slips. 

194.  A    Crossing-slip  is  an  arrangement  of  switch-rails,  in 
connection  with  a  set  of  crossing-frogs,  to  connect  two  trucks 
intersecting  at  a  small  angle. 

195.  Given    the    Angle    of    Intersection    of    Two    Straight 
Tracks,  to  Find  the  Length  and  Radii  of  Curvature  of  Slip-rails. 

In  Fig.  97  determine  EA  and 
AB  by  191 ;  then  assume  GE  or  BH 
(according  as  EA  is  less  or  greater 
than  AB)  as  small  as  the  crossing- 
frogs  will  permit.  Draw  the  radii 
HO  and  GO;  AH  =  AG  =  k  is  the 
known  tangent  for  the  central  angle 
F.  Hence 


=  ^l//cot  ^F  =  2kW,    (302) 
OL  =  R-$g  =  2klT-ff.     . 

For  the  theoretical  length  of  rails, 


(308) 


X  -— .  .    .     (304) 


F° 
57.3' 


(305) 


196.  Given  the  Angle  of  Intersection  made  by  the  Center 
Lines  of  a  Straight  and  a  Curved  Track,  to  Find  the  Radii  and 
Length  of  Slip-rails. 

FIKST  CASE. — Slip-rails  inside  main  curve. 

In  Fig.  98  determine  the  angles  F  and  Fl  at  B  and  Cby  192. 
Then  assume  KG  as  small  as  constructive  reasons  will  permit. 
Now 


(306) 


b  =  BON  =  (F,  -  F)  -  KOC,      .     .     .     (307) 
c  =  K OtH  =  F  +  b  =  F,  -  KOC.  (308) 


FROGS   AND   SWITCHES. 
From  129,  formula  (100), 


173 


(309) 


FIG.  98. 
For  the  lengths  of  slip-rails, 

UK  =  (Rl  + 

EL  =  (R\  — 


X 


c° 
57.3 


X  . 


.     (310) 


(311) 


SECOND  CASE.—  Slip-rails  outside  main  curve. 

Find  the  angle  F?  at  S,  F2  at  G,  and  OS  by  the  methods  of 
192.  Assume  (7  .2V  as  small  as  constructive  reasons  permit,  and 
calculate  angle  GON,  as  in  (306)  Then  NOS  =  F?  -  F3  -  QON, 
d  =  F3  +  GON  -  F*  -  NOS.  By  129,  formula  (106), 


The  remainder  of  the  solution  is  similar  to  the  first  case. 

197.  Given  the  Angle  of  Intersection  between  the  Center 
Lines  of  Two  Curves,  to  Find  Radii  and  Length  of  Slip-rails. 

FIRST  CASE  —  Slip-rails  on  concave  side  of  curves. 

In  Fig.  99  take  LG  as  small  as  constructive  reasons  permit. 
Join  L  with  0  ;  ihen 


174     A    FIELD-MANUAL    FOU    RAILROAD    ENGINEERS. 

Determine  angles  C00t,   C0,0,  and  side  00,  by  193.     Make 
LM  =  K0>  ;  then. 


and 


MO  =  (R  +  lg)  - 
M001  =  LOC  - 


=  R  -  Rl3 


FIG.  99. 


In  triangle  MOOi  two  sides  and  the  included  angle  are  now 
known,  and  the  triangle  m&y  be  solved.  02  is  the  center  of 
slip-rail  curves. 

OsMOi  =  020,3/  =  MOOi  +  MO.O, 


and 


*0i  =  180  - 


From  the  isosceles  triangle  GiO^M,  in  which  OiMaud  the  three 
angles  are  known, 

JfO, 


JfO,  = 


(314) 


Then 


R,  +  {g=  0,L  =  R,  -f  \(j  -  MO*,  .     .     .     (315) 
R*  -  lg  =  0*H  =:  &  -  &  -  MO*.  .     .     .     (316) 

The  central  angle  KO-.L  =  M0^0l  being  known,  GH  and  KL 
mav  be  found  as  in  196. 


FROGS    AND    SWITCH  Kri.  175 

SECOND  CASK. — Slip-rails  on  convex  side  of  curves. 

Let  the  (lotted  lines  of  Fig.  99  represent  this  case.  Assume 
AQ  and  compute  angle  AOQ  ;  produce  OQ  to  0-2',  the  center  of 
slip-mil  curve  ;  make  O^N  —  OJ Ol .  Reasoning  as  before,  find 
0,'N  =  Oa'Oi,  after  which  0/8,  02'P,  and  the  lengths  of  TS  and 
QP  may  be  found  as  in  the  first  case. 

Should  the  curves  intersect  as  in  Fig.  96,  no  difficulty  will  be 
found  in  computing  the  radii  and  length  of  slip-rails  by  follow- 
ing the  methods  used  above. 

These  methods  furnish  the  theoretical  length  of  slip-rails  ;  but 
us  the  theoretical  and  physical  switch-points  do  not  coincide,  the 
actual  length  will  be  considerably  less. 


CHAPTER  VI. 

CONSTRUCTION. 

ARTICLE  17.    DEFINITIONS;  GENERAL  CONSIDERATIONS;  VER- 
TICAL CURVES  ;  SUPERELEVATION  OF  OUTER  RAIL.  ' 

198.  The  work  of  locating  the  center  line  having  been  com- 
pleted, the  field  corps  is  usually  disbanded  and  a  new  one  organ- 
ized.    The  Chief  Engineer  still  remains  in  charge,  directing  the 
work  of  construction,  passing  on  bids  and  estimates,  arranging 
contracts,  and  attending  to  such  matters  of  importance  as  his  as- 
sistants are  unprepared  or  unauthorized  to  settle. 

199.  A  Division  Engineer  is  placed  in  charge  of  a  considerable 
length  of  line,  made  up  of  several  residencies.     To  him  the  resi- 
dent engineers  make  reports,  and  from  him  receive  directions 
and  orders  relating  to  construction.     These  reports  will  include 
monthly  estimates,  which  are  forwarded  to  the  chief  engineer  for 
inspection  and   approval.     Pay-rolls  for  the  men  employed  are 
made  out  in  the  office  of  the  division  engineer,  and  forwarded  to 
the  chief. 

200.  A  Resident  Engineer  is  placed  in  charge  of  a  few  miles  of 
line,  called  a  Residency,  and  has  direct  charge  of  the  construc- 
tion.    He  should  have  at  least  two  assistants— a  rod  man  and  an 
axeman — and  it   will   be   true   economy  to  allow   him  also  an 
assistant  who  can  take  his  place  at  the  instrument  and  assist  in 
superintending  construction. 

The  resident  engineer  is  usually  required  to  set  slope-stakes, 
locate  trestles  and  other  bridges,  tunnels,  culverts,  crossings,  and 
other  features  preceding  track-laying,  and  to  make  all  measure- 
ments upon  which  estimates  are  based  in  determining  the  com- 
pensation of  the  contractor. 

176 


CONSTRUCTION".  17? 

201.  The  Grade-line  is  determined  from  the  profile,  by  stretch- 
ing a  fine  thread  along  the  paper  and  so  adjusting  its  position 
that  the  proper  relation  between  cut  and  fill  is  obtained,  at  the 
same  time  that  the  maximum  gradient  is  not  exceeded.     The  cuts 
and  fills  should  be  made  as  small  as  possible,  at  the  same  time  that 
badly  broken  or  chopped  grades  are  avoided. 

The  Gradient  is  the  rate  of  change  of  elevation  of  grade-line, 
and  is  usually  expressed  in  per  cents,  a  1.2$  gradient  indicating  a 
rise  or  fall  of  1.3  feet  in  100  feet  horizontal.  When  the  grade  is 
ascending  the  gradient  is  marked  plus,  and  when  descending 
minus. 

The  word  grade  is  frequently  used  instead  of  gradient. 

The  grade-line  should  be  drawn  on  the  profile  in  red  ink,  with 
the  points  of  change  marked  by  a  cross  or  circle,  also  red.  The 
elevations  of  these  points  and  the  gradients  should  be  written  in 
red  above,  or  below,  the  grade  and  surface  lines. 

The  nature  of  the  work  and  the  disposition  of  material  from 
excavations,  and  the  availability  of  outside  material  for  embank- 
ments will  determine  whether  or  not  the  cuts  and  fills  must  balance. 
If  this  would  necessitate  a  long  haul,  it  will  often  be  preferable 
to  waste  material  from  excavation  and  borrow  for  embankments. 

A  Borrow-pit  is  an  excavation,  adjacent  to  the  line,  from  which 
material  is  taken  to  construct  an  embankment.  It  should  be 
separated  from  the  foot  of  the  embankment  by  a  space  termed 
a  Berm,  which  should  increase  with  the  height  of  embankment, 
never  falling  below  a  certain  minimum  width,  say  six  feet. 
Borrow-pits  should  be  regular  in  form,  with  sloping  sides  and 
drained  so  as  to  prevent  water  standing  in  them. 

202.  A  Cross-section  is  a  transverse  section  taken  at  each  sta- 
tion, and  at  intermediate  points  where   the  longitudinal  slope 
changes  considerably,  the  surface  between  adjacent  cross-sections 
being  approximately  such  as  would  be  generated  by  a  straight 
line  moving  on  these  end  sections  as  directors. 

203.  Slope-stakes  are  set  at  stations,  to  mark  the  points  on  cross- 
sections  where   the   side   slope  meets  the   ground-surface.     On 
them  the  cuts  or  fills  are  marked  on  the  niside,  while  the  outsides 
bear  the  station -numbers. 

No  slope-stakes  are  set  at  the  pluses  where  cross-sections  are 
taken,  unless  at  the  top  or  foot  of  bank  where  an  opening  is  left 
for  a  bridire  or  culvert. 


178     A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

The  notes  are  recorded,  however,  in  order  that  the  contents 
rnaj  be  correctly  calculated. 

204.  A  Grade-point  is  a  point  on  the  intersection  of  the  plane 
of  the  road-bed  with  the  ground-surface.     If  the  ground  is  level 
transversely,  a  single  stake  at  the  center,  marked  0  0,  will  suffice 
to  locate  the  point  of  passage  from  cut  to  fill.     When  the  ground 
is  not  level  transversely,  the  line  of  intersection  will  be  oblique  to 
the  axis  of  the  road  and  three  grade-stakes  are  needed,  one  at  the 
center  and  one  at  each  side. 

If  the  width  of  road-bed  in  excavation  differs  from  the  widlh  in 
embankment,  the  stake  should  be  set  at  the  edge  of  the  widest 
base. 

205.  To  Find   the   Grade-point   when  the   Ground   Slopes 
Uniformly  between  Stations. 


In  Fig.  100  let  AB  be  the  ground-line,  TO  the  grade-line,  and 
E  the  grade-point.  The  horizontal  distance,  x,  from  A  to  E  is 
required.  Let  the  cut  at^l  be  k>  ,  the  fill  at  B,  h* ,  and  the  length 
of  prism oid  I.  Draw  BG  parallel  to  CF.  From  the  similar  tri- 
angles AJSFsaid  ABG 


X  = 


(317) 


If  the  ground  does  not  slope  uniformly,  the  point  E  must  be 
found  by  trial,  such  that  the  rod-reading  equals  the  difference 
between  height  of  instilment  and  elevation  of  grade. 

206.  Vertical  Curves.— The  angle  formed  by  the  junction  of 
two  grade-lines  should  be  rounded  off  either  by  substituting 
several  small  changes  for  the  one  large  one,  or,  preferably,  by  in- 


CONSTRUCTION.  1<9 

sorting  a  regular  curve.  Where  the  algebraic  difference  of  gradi- 
ents is  less  than  0.3^  no  curve  will  be  needed,  while  for  larger 
differences  the  length  of  vertical  curve  should  vary  with  that 
difference,  unless  the  circumstances  of  the  case — such  as  the 
proximity  of  other  vertical  curves,  or  a  bridge— should  prescribe 
its  length.  In  any  case  the  length  may  be  either  assumed,  or  a 
given  rate  of  change  per  station  fixed  upon  and  the  length  com- 
puted. 

The  parabola  is  especially  well  adapted  for  vertical  curves,  be- 
cause of  the  ease  with  which  any  correction  may  be  found  when 
one  is  known,  since,  as  W7ill  presently  be  shown,  the  corrections 
vary  as  the  square  of  the  distance  from  the  point  of  tangency. 
A  second  property  of  this  curve  enables  us  readily  to  find  the 
correction  at  the  vertex,  or  meeting-point  of  grade-lines. 


FIG.  101 

In  Fig.  101  let  AC  and  CB  be  the  intersecting  grade-lines,  and 
AFB  tiie  curve  substituted  for  them.  Produce  AC  to  E  to 
meet  a  vertical  through  B.  Draw  the  vertical  (7(7.  Then  will 
CF  =  FG  —  m  by  the  second  property  referred  to.  Since 
measurements  are  made  horizontally,  the  similar  figures  AGO 
and  AEB  furnish  the  relation  OF  =  \CG  =%EB.  Calling  the 
algebraic  difference  of  gradients  d,  and  the  length  of  curve  21, 


(a) 


If  the  rate  of  change  of  gradient  per  station  be  a,  it  is  evident 
that 


The  equation  of  the  parabola  referred  to  A  as  origin  may  be 
written 

in 
fa 


189     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

To  find  the  correction  UK  —  z  at  a  distance  x  from  A,  we  have 
from  the  similar  triangles  AHL  and  ACO, 

HL  =  6'O-|  =  2mj (d) 

But  KL  —  y,  and  z  —  HL  —  KL,  or,  inserting  values, 
_  2inx  _  /2mx 

"        r  '~ 


or  z  =  m—  ..........     (318) 

Insert  the  value  of  d  from  (&)  in  (a),  and  the  resultant  value  of 
m  in  (318);  then 


When  x  =  1  station,  z,  —  i«  -.  when  x  =  2  stations,  22  —  2a,  etc. 

It  will  only  be  necessary  to  figure  corrections  for  one-half  the 
curve,  as  they  are  the  same  for  corresponding  points  each  side  of 
the  vertex.  If  preferred,  however,  ail  corrections  may  be  com- 
puted from  the  first  tangent  produced. 

EXAMPLE.—  A  -f  0.9$  meets  a  —  0.6#  grade  at  sta.  181,  the  ele- 
vation of  which  is  91.0ft.  Required  the  corrections,  and  cor- 
rected grade  elevations  for  points  100  ft.  apart. 

Here  the  algebraic  difference  of  gradients  is  0.9  —  (—  0.6)  =  1.5 
Suppose  a  be  taken  as  0.25,  or  the  length  of  curve  as  6  stations. 

Formula  (a)  gives  m  =  %  X  3  X  1.5  =  1.125  feet. 

At  the  P.O.,  sta.  178,  z  —  0;  at  179,  (318)  or  (318')  gives  z,  = 
0.125;  at  180,  2,  =  4  X  0.125  =  .50.  The  original  and  corrected 
grade  elevations  are  as  follows: 

Sta.         178    179     180     181     182     183     184 


Original  elevation. 

88.3 

89.2 

90.1 

91 

.0 

90 

.4 

80 

.8 

89 

2 

Corrections  

0.0 

0.125 

0.50 

1 

125 

0. 

50 

0 

.125 

U 

0 

Corrected  elevat'u 

88.30 

89.075 

89.60 

89. 

875 

89. 

90 

68. 

675 

89 

20 

*  If  a  circle  be  taken  as  the  joining1  curve  we  may  derive  (318')  by  finding 
R  in  terras  of  a,  theu  writing  D  =  5730  -+-  R,  and  n  =  x,  in  formula  (3G). 


CONSTRUCTION. 


181 


EXAMPLE   2.—  A   +  0.3#    meets  a  -f  1.1*   grade  at    sta.    312, 
whose  elevation  is  155  0      Find  corrections. 


Take  a  =  0.2  in  this  case;  then  I  = 


=  2  sta- 


lions.     The  corrected  grade  heights,  etc.,  will  be  as  follows  : 

Sta.  310  311  312  313  314 


Original  elevation 154.4        154.7        155.0 

Corrections 0.0  0.1  0.4 

Corrected  elevation 154.4        154.8        155.4 


156.1  157.2 
0.1  0.0 

156.2  157.2 


The  reason  for  adding  the  corrections  in  this  case  will  be  evi- 
dent from  a  figure. 

The  table  below  gives  the  corrections  in  feet,  for  certain  alge- 
braic differences  of  gradients  and  lengths  of  curve,  at  intervals  of 
50  ft.  each  way  from  the  vertex.  When  the  difference  of 
gradients  is  plus,  the  correction  must  be  subtracted  from  the 
original  grade  elevation  ;  when  the  difference  is  minus,  the  cor- 

TABLE    OF   CORRECTIONS    FOR   VERTICAL   CURVES. 


jebraic 
ifference 
°  Gradi- 
its. 

•ol|| 

Distance  from  Vertex  in  Feet. 

<; 

rfoocfi 

0 

50 

100 

150 

200 

250 

300 

350 

400 

0.3 

0.075 

0.15 

0.08 

0.04 

0.01 

0 

0  4 

.10 

.20 

.11 

.05 

.01 

0 

0.5 

.125 

.25 

.14 

.06 

.02 

0 

06 

.15 

.30 

.17 

.08 

.02 

0 

0.7 

.175 

.35 

.20 

.09 

.02 

0 

0  8 

.20 

.40 

.23 

.10 

.03 

0 

0.9 

.225 

.45 

.25 

.11 

.03 

0 

1.0 

.25 

.50 

.28 

.13 

.03 

0 

.1 

.1833 

.83 

.57 

.37 

.21 

.09 

.02 

0 

.2 

.20 

.90 

.63 

.40 

.23 

.10 

.03 

0 

.3 

.2167 

.98 

.68 

.44 

.24 

.11 

.03 

0 

.4 

.2333 

1.05 

.73 

.47 

.26 

.12 

.03 

0 

.5 

.25 

1.13 

.78 

.50 

.28 

.13 

.03 

0 

1.6 

.2667 

1.20 

.83 

.53 

.30 

.13 

.03 

0 

1.7 

.2833 

1.28 

.89 

.57 

.32 

.14 

.04 

0 

1.8 

.30 

1.35 

.94 

.60 

.34 

.15 

.04 

0 

1.9 

.2375 

1.90 

1.46 

.07 

.74 

.48 

.27 

.12 

.03 

0 

2.0 

.25 

2.00 

1.53 

.13 

.78 

.50 

.28 

.13 

.03 

0 

2.1 

.2625 

2.10 

1.61 

.18 

.82 

.53 

.30 

.13 

.03 

0 

2.2 

.275 

2.20 

1.68 

.24 

.86 

.55 

.31 

.14 

.03 

0 

2.3 

.2875 

2.30 

1.76 

29 

.90 

.58 

.32 

.14 

.04 

0 

2.4 

.30 

2.40 

1.84 

.35 

.94 

.60 

.34 

.15 

.04 

0 

2.5 

.3125 

2.50 

1.91 

.41 

.97 

.63 

.35 

.16 

.04 

0 

2.6 

.325 

2.60 

1.99 

.46 

1.02 

.65 

.37 

.16 

.04 

0 

182     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

rection  must  be  added.  Similar  tables  for  other  lengths  of  curve 
or  differences  of  gradients  may  be  computed  by  the  engineer,  and 
time  in  the  field  saved  by  their  use.  In  setting  grade-stakes,  it 
will  be  well  to  set  them  50  ft.  apart  on  vertical  curves,  though  to 
allow  for  the  vertical  curve  at  each  regular  station  will  suffice 
when  cross-sectioning. 

207.  Elevation  of  Outer  Rail  on  Curves.— In  138  it  was 
shown  that  the  superelevation  of  outer  over  inner  rail  might, 
for  standard  gauge  track,  be  given,  nearly  enough,  by  the  formula 

F2 
*&SiB* (134) 


in  which  e  is  the  elevation  in  feet,  and  Fthe  velocity  in  miles  per 
hour.     Writing  R  —  ——-  in  this  formula  gives 

e  =  0  000058  F2Z> (319) 

The  following  table  has  been  computed  by  formula  (319). 


TABLE  OF  SUPERELEVATIONS  OF  OUTER  RAIL. 


F"in 

Degree  of  Curve. 

Milt-s 

per 

Hour. 

1° 

2° 

3° 

40 

5° 

6° 

7° 

8° 

9° 

10° 

12° 

20 

.02 

.05 

.07 

.09 

.12 

.14 

.16 

.19 

.21 

.23 

.28 

30 

.05 

.10 

.16 

.21 

.26 

.31 

.37 

.42 

.47 

.52 

.63 

40 

.09 

.19 

.28 

.37 

.46 

.56 

.65 

.74 

.84 

.93 

50 

.15 

.29 

.44 

.58 

.73 

.87 

1.02 

1.16 

CO 

.si 

.42 

.63 

.84 

1.04 

1.25 

Since  grade-stakes  are  set  at  the  edge  of  base  it  will  be  neces- 
sary to  determine  the  difference  between  these  elevations  and  the 
elevations  of  center  line.  Calling  this  difference  h,  the  half-base 
&,  and  the  distance  between  centers  of  rail-heads  for  standard 
gauge  4.9  feet,  we  shall  have,  from  similar  triangles, 

b 


4.9 


e  =  Q.Sbe  (nearly) (320) 


If  the  inner  rail  is  required  to  remain  at  grade  (320)  will  become 

h  =  0.2&?  ±0.5^, (320') 

according  as  the  grade-stake  is  to  be  set  at  outer  or  inner  edge  of 
base. 


CONSTRUCTION.  183 

EXAMPLE. — What  will  be  the  value  of  li  when  e  —  .46,  the 
base  being  14  feet? 

By  (320),  h  =  0.2  X  7  X  0.46  =  0.64  feet.  The  outside  is  this 
much  higher  than  the  center,  the  inside  edge  this  much  lower. 

The  superelevation  of  outer  rail  .should  be  computed  for  the 
highest  speed  at  which  trains  are  to  be  run  over  the  curve;  the 
maximum  iillowed  in  practice  rarely  exceeds  8  inches,  since  a 
greater  elevation  would  endanger  the  slow-running  freight  trains. 
Even  when  the  theoretical  superelevation  is  given  the  outer  rail, 
it  is  more  worn  than  the  inner  one,  either  because  there  are  other 
forces  acting,  or  because  of  the  sliding  action  of  the  outer  wheel 
due  to  imperfect  adjustment  where  the  original  coning  has  been 
destroyed  by  wrear. 

Engineers  sometimes  elevate  the  outer  rail  1  inch  per  degree  up 
to  3°,  and  make  e  =  3J  inches  for  a  4°  curve,  4  inches  for  a  5° 
curve,  and  4|  inches  for  a  6°  curve.     Still  other  rules  are  in  use. 

If  transition-curves  are  not  employed,  the  difference  of  eleva- 
tion is  the  same  from  P.  C.  to  P.  T.,  fading  out  to  nothing  on 
tangent.  The  elevation  begins  on  tangent  from  50  to  200  feet 
back  of  P. C.,  depending  on  the  amount  the  outer  rail  is  to  be 
raised. 

208.  Easing   Grades    on    Curves. — To    compensate   for  the 
increased  resistance  due  to  curvature,  it  is  customary  to  reduce 
the  grade  on  curves.     This  resistance  is  taken  to  vary  directly  as 
the  curvature;  a  rule  often  used  is  to  reduce  the  gradient  0.05 
foot  per  degree  of  curve 

ARTICLE  18.     EARTHWORK. 

A.   Setting  Slope-stakes 

209.  Slope-stakes  are  set  at  the  points  where  the  side  slopes 
meet  the  ground- surf  ace,  to  mark  the  limits  of  the  excavation  or 
embankment,  and  to  show  the  constructor  what  the  cut  or  fill 
must  be.     In  Fig.  102,  KAE  represents  the  ground-surface,  HBC 
the    grade-surface.     Let   AB  =  h   be    the    center   height.     Let 

EL 

—  s  be  the  side  slope,  which  varies  with  the  nature  of  the 

KL 

material;  for  earth-excavation  the  side  slope  will  average  about  1 

to  1,  so  that  s=l,  W7hile  for  ordinary  earth-embankment  it  will 


184     A    FIELD-MANUAL   FOR   RAILROAD    EXGIXEKRS. 

average  about  H  to  1,  so  that  s  =  1£.  The  side  height  for  level 
sectioos  is  the  same  as  the  center  and  may  be  found  for  any 
section,  so  that  the  distance  LH  is  required.  From  the  equation 


FIG.  102. 

defining  s,  EL  =  KLs  =  Tis.     Let  the  base  HC  =  26;    the  "  dis- 
tance out  "  from  center  is 


d.o  =  BL  = 


hs. 


210.  Surface  Inclined.  —  Where  the  ground  slopes  transversely 
the  position  of  the  slope-stake  cannot  be  found  from  the  center 
height  unless  the  slope  of  the  ground-surface,  as  well  as  the  side 
slope,  is  known.  The  slope-stakes  can  be  most  easily  and 
rapidly  set  by  trial. 


0       s 


FIG.  103. 

In  Fig.  103,  FAE  is  the  ground-surface,  AB  =  h  the  fill  at 
the  center.  We  have  to  find  the  distances  out  of  15  mud.  F  from 
A,  and  the  side  heights  ME—  hi  ,  and  NF '  =  h*.  Let  OP  repre- 
sent the  plane  of  the  instrument  at  a  height  II. I.  above  the 
datum,  obtained  from  the  known  elevation  of  a  bench  or  turn- 
ing-point. PB  is  the  height  of  the  plane  of  the  instrument  above 
the  grade.  Call  PB  the  Station  Constant  (s.c.). 


CONSTRUCTION".  185 

The  fill  at  A  will  evident1/  equal  the  rod  reading  less  the 
station  constant.  Mark  this  on  the  center  stake. 

Since  the  ground  slopes  downward  from  A  to  E,  the  distance 
out  will  be  greater  than  for  a  level  section,  while  for  F,  on  the 
higher  side,  it  will  be  less. 

Suppose  we  take  a  reading  QK  at  a  distance  out  =  b  -\-  lis  ;  the 
fill  at  that  point  is  LK  —  QK  —  QL  —  r  —  s.c.,  and  the  corre- 
sponding distance  out  is  d.o.  —  b  -f-  KL  X  *,  which  is  greater 
than  All,  since  LK  is  greater  than  h.  If  now  a  reading  is  taken 
at  the  distance  out  b  -\- <s  X  LK,  we  shall  have  a  fill  greater 
than  LK,  unless  the  ground  is  level  from  Kto  E,  and  therefore 
b  -)-  s  X  LK,  the  distance  out  actually  used,  will  be  less  than 
that  called  for  by  the  reading.  However  we  shall  have  obtained 
a  closer  approximation  to  the  position  of  E,  and  by  repetitious  of 
this  process  may  come  as  close  to  its  true  position  as  the  con- 
ditions require. 

The  same  thing  can  be  accomplished  more  rapidly  by  estimat- 
ing the  fall,  by  the  eye,  from  A  to  a  point  b  -f  his  out,  then  mul- 
tiply this  estimated  fall  by  the  slope  and  add  to  b  -\-  hs.  Take  a 
reading  r  at  this  distance  out;  then  compute  the  d.o.  for  the  fill 
r  —  s.c.  and  note  if  this  agrees  with  the  actual  d.o.  If  it  does 
not,  make  a  new  trial  with  this  reading  as  a  guide. 

For  ordinary  work  the  actual  and  computed  distance  out  should 
be  such  that  if  the  rod  were  held  at  the  computed  distance  the 
new  distance  would  not  differ  more  than  a  tenth  from  that  just 
computed.  The  stake  is  then  set  at  the  computed  distance  out. 
After  a  little  practice  it  will  be  found  that  the  second  setting  of 
the  rod  may  usually  be  made  to  fall  as  close  to  the  true  position 
as  the  limit  requires. 

When  the  stake  is  marked  and  driven  the  cut  or  fill  at  that 
point  and  the  distance  out  are  recorded  in  the  notes. 

As  an  example  suppose  26  =  14  feet,  s  =  li  (i.e.,  slope  1£  to  1), 
H.I.  —  187.3,  grade  elevation  =  184.0.  The  station  constant  is 
s.c.  —  187.3  -  184.0  =  3.3.  Suppose  the  rod  at  center  to  read 
8.5  ;  the  fill  will  be  8.5  —  3.3  =  5.2,  which  mark  on  stake  as 
"  F.  5.2. "  The  distance  out,  if  section  were  level,  would  be  d.o.  = 
7  +  5.2  X  1.5=  14.8;  but  suppose  the  ground  rises  and  we  esti- 
mate the  rise  as  1  foot,  which  multiplied  by  s  gives  1.5  feet  to  be 
subtracted  from  14.8,  since  this  is  on  the  higher  side  of  center 
for  a  section  in  embankment.  Let  the  reading  c,t  13  3  out  be 
7.7,  which  gives  a  fill  of  7.7  -  3.3  —  4.4  feet,  calling  for  a  d.o.  = 
74-4.4  X  1-5  =  13.6.  This  shows  we  are  too  far  in,  but  as  u 


186     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


reading  further  out  will  be  less,  giving  a  correspondingly 
smaller  d  o.,  we  try  a  reading  at  13.5  feet  out.  Suppose  the  read- 
ing to  be  7.6;  the  fill  will  be  7.G  -  3.3  =  4.3,  calling  for  a  distance 
out  of  13.45  feet,  which  agrees  almost  exactly  with  the  trial  dis- 
tance. The  stake  is  marked  "  F.  4.3,"  and  the  result  recorded 
in  the  cross-section  book. 

On  the  other  side  of  the  section  suppose  we  estimate  the  fall  to 
be  1.5  feet  in  15;  we  should  try  a  reading  at  13.8  +  1.5  X  1.5  =  10.1, 
say  16.0  feet.  Let  this  reading  be  9.0  ;  the  fill  will  be  9.0  -  3.3 
=  5.7  feet,  calling  for  a  d.o.  =  7  -f  5.7  X  1.5  =  15.6,  which  shows 
our  reading  was  taken  too  far  out.  Try  a  reading  at  15.4,  which 
suppose  8.9  ;  the  fill  is  8.9  —  3.3  =  5.6,  and  the  d.o.  =7+5.6 
X  1.5  =  15.  4,  which  agrees  exactly  with  the  trial  distance. 

In  excavation  the  method  of  proceeding  is  the  same  as  in  em- 
bankment, except  that  s  has  generally  a  different  value.  For  solid 
rock  s  is  usually  J,  that  is,  the  slope  is  taken  as  J  to  1;  for  loose 
rock,  gravel,  and  ordinary  earth  the  slope  may  be  taken  as  1  to  1. 

The  station  constant  in  cuts  is  always  positive,  and  the  rod 
reading  has  to  be  subtracted  from  it  to  obtain  the  out.  In  fills, 
when  the  ILL  is  greater  than  the  grade  height,  the  fill  equals  the 
difference  of  the  rod  reading  and  the  station  constant.  When 
ihe  H  L  is  less  than  the  grade  height  the  rod  reading  plus  the 
s.c.  gives  the  fill. 

211.  The  Notes  may  be  kept  in  the  form  below,  which  repre- 
sents one  page  of  the  cross-section  book.  The  cut  or  fill  is  written 
above  the  line,  the  distance  out  below.  A  plus  sign  indicates  a 
cut,  a  minus  sign  a  fill 


Sta. 

Ground. 

Grade. 

Left. 

Center. 

Right. 

161 

178.8 

184.0 

-  4.4 
13.15 

-  5.2 

-  5.6 

TsT 

0  8 

20 

162 

181.6 

183.0 

"sT 

-  1.4 

10.0 

+  20 

182.2  g 

0.0 
9.0    , 

-  O.G 

-  1.0 

8.5' 

+  48 

182.5  ' 

+  0.9 

9.9^ 

0.0 

-  0.4 
7.6 

+  66 

183.5 

+  2.4 

+  1.2 

0.0 

11.4 

9.0 

163 

185.0 

182.  0 

+  4.4  +2.8 

+  3.0 

+  4.3  +  S.« 

o.i      11.  <; 

13.4      10.0 

CONSTRUCTION.  187 

212.  Irregular  Sections. — When   readings  are  taken  only  at 
the  center  and  sides  it  is  termed  a  "three-level  section."     Very 
irregular  ground  may  require  several  more  readings  iu  order  to 
determine  its  area  ;   in  this  case  a  reading  is  taken  at  each  change 
of  surface  in  the  section,  and  the  cut  or  fill,  together  with  the  dis- 
tance out,  recorded— the  distance  being  measured  from  the  center 
to  the  point  where  the  rod  was  held  in  taking  the  reading. 

When  the  base  cuts  the  ground -surf  ace  the  section  is  partly  in 
excavation  and  partly  in  embankment,  but  each  side  will  be 
staked  out  in  the  manner  described  above.  The  distance  of 
grade-point  from  center  must  be  found  and  recorded. 

213.  Staking  Out  Openings. — Where  openings  are  to  be  left 
for  trestles,  culverts,  and  other  structures,  stakes  must  be  set  to 
mark  the  limits  of  the  embankment.     Stakes  marked  T.  B.  are 
set  at  the  center  and  sides  to  fix  the  place  where  the  top  of  bank 
is  to  end  ;  other  stakes,  marked  F.  S.,  are  set  at  the  foot  of  slope, 
the  plus  at  which  they  fall — together  with  the  distance  out  from 
center— being  recorded  in  the  note-book.     The  slope  of  the  toe 
of  dump  should  be  the  same  as  the  side  slope. 

214.  Marking  Stakes.— All  slope  and   toe  stakes   that  limit 
excavation  or  embankment  should  be  driven  with  tops  inclined 
outward  from  the  center.     The  cut  or  fill  is  marked  on  inside  iu 
plain  figures  preceded  by  the  letter  C.    or  F.  as  being  more 
easily  understood  by  the  contractor  than  the  plus  and   minus 
signs  used  in  the  notes.     The  reverse  side  should  bear  the  station 
number. 

215.  Shrinkage — Growth. — It  must  be  remembered  that  earth- 
work in  embankment  will  settle,  or  shrink  in  volume,  even  after 
having  been  compacted  by  the  feet  of  the  teams  during  construc- 
tion.    Where  the  fill  is  not  great,  allowance  may  be  made  for 
shrinkage  when  setting  grade-stakes,  but  in  heavy  fills  allowance 
should  be  made  when  the  stakes  are  set  for  construction.     The 
proper  allowance  will  vary  with  the  nature  of  the  material,  but 
about  10  per  cent  will  be  a  fair  average.     The  contract  should 
always  specify  the  amount  of  shrinkage  to  be  allowed  on  par- 
ticular works.     If  the  earth  is  measured  in  the  borrow- pits,  an 
equivalent  allowance  should  be  made,  since  earth  is  more  com- 
pact in  embankment  than  before  excavating. 

With  rock,   however,  it  is  found  that  the  volume  increases 


188     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

after  excavation,  and  this  increase  is  termed  growth.  The  size  of 
the  fragments  will  determine  the  growth,  which  will  vary  from 
one  half  to  five  eighths  of  the  original  volume — the  larger  the 
fragments  the  greater  the  increase.  Little  or  no  allowance  need 
be  made  for  settlement  when  placed  in  embankment. 

216.  Borrow-pits  should  be  regular  in  form,  particularly  if 
the  volume  of  earth  moved  is  to  be  measured  in  the  borrow-pit. 
They  should  be  properly  drained  to  prevent  water  standing  in 
them  and  should  have  an  ample  berm  between  edge  of  pit  and 
foot  of  slope,  the  width  of  berm  increasing  with  the  height  of 
embankment. 


B.  Areas  of  Sections. 

217.  Before  the  volume  of  earth  in  excavation  or  embank- 
ment can  be  computed  the  area  of  each  cross-section  must  be 
found.     To  do  this  divide  the  section  into  triangles  and  trape- 
zoids,  find  the  area  of  each  separately,  and  take  the  sum.     To 
shorten  the  calculations  a  few  simple  rules  will  be  deduced. 

When  the  center  and  side  heights  are  equal  we  have  a  one- 
level  section;  when  the  center  and  side  heights  differ  it  is  a 
three-level  section ;  where  the  height  is  found  at  five  places  in 
the  section  it  is  a  five-level  section ;  and  so  on. 

218.  To  Find  the  Area  of  a  Three-level  Section. 


Fia.  104. 


In  Fig.  104  the  area  ABGDF  is  required.  Draw  EA  and  EG, 
dividing  the  area  into  four  triangles.  With  the  notation  of  the 
figure  it  is  seen  that  triangles  (1)  and  (2)  have  equal  bases  b  and 


CONSTKlTCTlOtf.  189 

altitudes  J/i  and  hi  ,  while  (3)  and  (4)  have  the  common  base  h$ 
and  altitudes  Dl  =  b  -f  Hi  s  and  dl  —  b  +  h^s.     By  geometry, 


Area  (3)  +  (4)  =  =  * 

The  area  of  the  whole  section  is  therefore 


•A  =  ,       +  *_      b      +  k..  (8S1) 

This  formula  affords  an  easy  method  of  obtaining  the  area  of  a 
three-level  section,  and  when  written  in  words  becomes  the  follow- 
ing 

RULE.  —  Multiply  the  half-sum  of  the  side  heights  by  the  half-base 
and  to  this  add  the  product  of  the  center  height  by  the  half  -sum  of 
the  distances  out;  the  result  will  be  the  area. 

If  the  linear  measurements  are  in  feet,  the  area  will  be  in 
square  feet. 

When  //,  =  h,  =  7i0  ,  then  Dl  =  dl  =  d  =  b  +  7t0s  and  (321)  re- 
duces to 

A  =  bh0  +  h»d=  /<„(&  +  d)  =  7/0(2b  +  7i0«).     .     .     (322) 

The  section  is  now  a  trapezoid  or  one  level  section  for  which 
(322)  may  be  deduced  by  the  usual  rule  for  the  area  of  a  trapezoid. 

219.  Area  of  Five-level  Section. 

Fig.  105  represents  this  case,  where  we  may  evidently  divide 


i                            I 
r\_ y,^ ._        .A._. »^l 


FIG.  105. 


the  area  into  triangles  and  trapezoids,  computing  the  area  of 
eacli  separately  and  taking  their  sum  for  the  whole  area.  The 
following  simpler  method  may  be  preferred: 


100     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

Write  the  notes  as  in  field-book,  except  that  center  height  is 

0 
6 


placed  over  zero  and  an  additional  -  is  written  at  each  end  as 


below. 


Beginning  at  the  center,  multiply  heights  by  distances  out  in 
pairs  as  indicated  by  the  sloping  lines,  the  products  of  members 
connected  by  full  lines  being  plus  and  of  those  connected  by 
dotted  lines  minus.  Half  the  sum  of  the  products  will  be  the 
area,  thus: 


_  _  DJL,  +  0 

=  dji,  +  0 

The  grouping  is  symmetrical  for  areas  each  side  of  center,  as 
(323)  exhibits;  so  it  will  be  sufficient  to  show  that  the  rule  is  cor- 
rect for  either  side.  Divide  the  figure  into  trapezoids  and  tri- 
angles as  shown;  then 

Area  trapezoid  BCKM  =  $(7*0  +  Hi)Dl  , 

ABMN  =  $(#,  +  Hz)(D,  -  -Di), 
"     triangle  ALN       =  |#2(Z>2  -  b). 


The  two  trapezoids  include  the  triangle  ;  hence  the  latter  must 
be  subtracted  from  their  sum.  Doing  this  and  simplifying,  we 
have 

AL  =  l(h0D,  +  SiD*  -  DiH*  +  HJ>), 

which  is  the  same  as  results  in  (323). 

220.  General  Formula  for  Areas.  —  The  method  of  219  may 
be  applied  to  any  section  no  matter  how  irregular.  Suppose 
there  have  been  n  levels  taken  on  one  side  exclusive  of  the  center 
height;  the  notes  would  appear  as  below: 


/»,  fln-i       7i 


^ 
0      di       da          '   dn-i      d^      b 

Expanding  in  the  same  manner  as  in  219, 

R  =i[(7*odi+7t1d9-f-    .  .  7iB_1dn+74M&)-(<Ua+  .  .  .  dn-M].  (324) 

To  show  that  (324)  gives  the  true  area,  consider  that  we  have  n 


CONSTRUCTION".  191 

trapezoids  whose  area  is  positive,  and  one  triangle  whose  area  is 
negative  and  equal  to  i  7in(dn  —  b). 
Writing  out  the  area,  we  have 

AR  =  4[(7<o  -f  7i,)di  +  (74l  -f  7ta)(da  -  d,) 

+  .  .   .  (7iB_i  +  7ln)(dn  -  d»-l)  -  hn(dn  -  b)]. 

Perforraiug  the  indicated  operations  and  simplifying, 


\vhich  is  the  same  result  obtained  in  (324). 

Evidently  n  may  have  any  positive  integral  value. 

If  preferred,  the  cross-sections  may  be  plotted  on  cross-section 
paper  and  the  area  read  off  by  means  of  a  planimeter. 

221.  Tables  of  Areas  of  Level  Sections,  and  the  Three- 
level  Correction.  —  Formula  (322)  may  be  employed  in  com 
putiug  the  areas  of  level  sections  for  any  values  of  b  and  s. 

Table  XVII  gives  the  areas  for  a  few  of  these  values.  When 
many  sections  are  to  be  figured  it  will  be  well  for  the  engineer  to 
compute  the  necessary  tables,  provided  he  is  unable  to  secure 
published  ones  for  the  particular  bases  and  slopes  he  is  working 
with.  It  is  not  within  the  scope  of  this  volume  to  give  the 
variety  of  tables  needed;  they  are  published  elsewhere. 

The  area  of  three-level  sections  may  be  found  from  the  areas  of 
level  sections  by  the  aid  of  a  suitable  correction.  Let  the  height 
used  in  entering  the  tables  of  level  sections  be  the  mean  height  of 

the  three-level   section,  lim  —  —  —  ;  the  corresponding 

area,  by  (322),  is 

A'  =  7im(2b  +  hms)  =  2b7im  +  h^s  ........     (a) 

The  true  area  is  given  by  (321): 

A  =  >*  +  »!  f  ».&  +  *.*+_^ 

=  „&  +  *£  ±A<  +  8A/A  +  «i±*:.  _  *.,. 

4  4 

=  2b7im  -f  2A07*ms  —  7i02s  ..........     (b) 

From  equations  (a)  and  (b)  the  correction  is 

e  =  A'  -  A  =  (7tm*  -  2h0hm  +  7i0a)«  =  (7im  -  7u)2*.    (325) 


192     A   FIELD-MANUAL  FOR  RAILROAD   ENGINEERS. 

Table  XVIII  was  computed  by  (325),  and  gives  values  of  c 
which  are  always  positive,  and  which  must  be  subtracted  from 
the  tabular  area,  found  by  entering  the  table  with  the  mea?i 
height  of  section,  in  order  to  get  the  true  area. 

EXAMPLE.— The  side  heights  for  a  till  having  a  14  ft  base  and 
side  slopes  1£  to  1  are  8.6  and  16.4  feet,  while  the  center  height  is 
7.7  ft.  The  mean  height  is 

8.6  +  2X7.7  +  16-4 

lim  — =  10.1  it., 

4 

for  which  Table  XVII  gives  A'  =  294.4  sq.  ft.     For  the  correc- 
tion, lim  -  h0  =  10.1  -  7.7  =  2.4  ft.,  for  which   Table   XVIII 
gives  c  =  8.6  sq.  ft. 
The  true  area  is  now  A  =  294,4  —  8.6  =  285.8  square  feet. 


C.   Volume  of  Earthwork. 

222.  Cross-sections  must  be  taken  at  all  full  stations  and  at  in- 
termediate points,  or  pluses,  where  there  is  a  change  in  longitudi- 
nal slope.     It  will  be  well  in  any  event  to  take  them   so  close 
together  that  the  difference  in  end  heights  should  not  exceed 
about  five  feet.    The  time  consumed  in  making  these  intermediate 
measurements  will  be  more  than  offset  by  the  reliability  of  the 
results.     A  few  of  the  more  usual  methods  of  estimating  quan- 
tities will  be  given  here. 

223.  Averaging  End  Areas.  —  This  is  the  easiest  of  application 
and  therefore  the  most  generally  used,  but  is  open  to  the  objec- 
tion that  it  gives  inaccurate  results.     However,  when  bids  are 
based  upon   it,  both  parties  to  the  contract  agreeing,  it  would 
seem  to  answer  as  well  as  any  other  method. 

If  Ai  and  A9  are  the  end  areas  and  I  the  length,  we  shall  have 


(326) 


Stated  in  words,  (326)  yields  the  following 

RULE.  —  Multiply  the  half-sum  of  the  end  areas  by  the  axial 
length  of  prumoid  ;  the  result  will  be  the  volume. 

If  areas  arc  in  square  feet  and  length  in  feet,  the  volume  will 
be  in  cubic  feet  ;  to  reduce  to  cubic  yards  divide  by  27. 


CONSTRUCTION.  193 

224.  Prismoidal  Formula. — The  parallel  sections  should  be 
so  taken  that  the  surface  bounding  the  volume  to  be  measured 
may  be  supposed  to  be  generated  by  a  straight  line  moving  on 
the  bounding  lines  of  the  sections  as  directors  in  such  a  manner 
as  to  return  to  its  original  position.  Such  a  figure  is  called  a 
prismoid. 

The  height  of  anyv  section  intermediate  between  the  end 
sections  is  a  function  of  its  distance  from  either  end,  and  the 
area  of  that  section  will  be  a  quadratic  function  of  its  distance 
from  either  end. 

Now  we  know  from  mechanics  that  Simpson's  (Newton's)  Rule 
will  hold  for  any  function  not  higher  than  the  third  ;  so  for  the 
mean  area  this  rule  yields 

-».-                     AI-$-  4:Am  ~f-  A? 
Mean  area  = — 


where  Al  and  J.2  are  tke  end  areas,  Am  the  middle  area ;  hence 
we  have  for  the  volume 

V  =  (At  +  4Am  +  A*)±,        .     .     .     (327) 

in  which  I  is  the  axial  length  of  prismoid. 

The  same  result  may  be  obtained  geometrically  by  dividing  the 
prismoid  into  prisms,  wedges,  and  pyramids,  and  applying  the 
usual  rule  for  volumes. 

Let  the  end  areas  be  a1}  a2,  a/,  a2',  etc.,  and  the  mid-areas 
am,  am',  etc. 

For  the  prism  «i  =  «2  =  am ;  hence  the  volume  is 

v  =  ail  =  (ch  -f  4«,n  +  aa)— (a) 

o 

For  the  wedge  «/  —  2am'  and  aa'  =0;  therefore 

v'  —  ai  TT  —  (a\  ~h  4«m'  -j-  a^')—.      ...     (5) 
2  0 

For  the  pyramids  a/'  =  4«w"  and  a*"  =  0;  hence 

I  I 

V"  =  tf'—  =  (a/'  ^  iam"  -f  a,")g-,       .     .     (o) 


194     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

Adding  (a),  (6),  and  (c),  the  total  volume  is 
V  =  v  +«'  +  v" 

=  [(ai+a1f+a1r')+4(am^am'+am'')+(az+av'+az'')]'Q.    .     (d) 
But  «i  -\-  «/  +  ai"  =  4i 

«m  +  «m'  +  «>n"  =  -4m, 

and  #2  -f-  #2'  -f~  ^2"  =  AI  ; 

therefore  F=  (Ai  -f  ±Am  +  A)  •  -«, 

o 

the  same  as  (327). 

Slated  in  words  there  results  the  following 

RULE. — To  the  sum  of  the  end  areas  add  four  times  the  mid-area, 
multiply  by  the  length,  and  divide  by  6.  The  result  will  be  the 
volume. 

To  reduce  to  cubic  yards,  divide  by  27. 

Formula  (327)  contains  three  terms,  the  middle  area  being 
derived  from  the  cross-section  notes  for  the  end  sections  at  the 
expense  of  some  little  trouble.  In  the  attempt  to  simplify  this 
formula  Dr.  George  Bruce  Halsted  in  1881  published  a  two-term 
prismoidal  formula,  giving  the  volume  in  terms  of  one  base  and 
a  section  at  two  thirds  of  the  length  of  the  prismoid,  the  formula 
being 

V=(A>  +  34s)  £  =  (34  j  +  4',)|.   .     .     .     (328) 

In  1894  Professor  W.  II.  Echols  showed  by  the  aid  of  higher 
mathematics  that  an  indefinite  number  of  two-term  formulae 
might  be  derived.  The  same  results  were  established  in  1895  by 
Professor  T.  U.  Taylor  by  elementary  mathematics. 

None  of  these  two-term  formulae  have  so  far  been  placed  in  a 
form  suitable  for  application  to  earthwork  measurement,  owing 
to  the  difficulty  of  finding  the  area  of  the  auxiliary  section. 

In  fact  the  only  objection  to  the  use  of  (327)  is  the  loss  of  time 
required  in  obtaining  the  mid-area  and  the  uncertainty  as  to  its 
accuracy  in  the  case  of  very  irregular  sections. 

For  three-level  ground  we  may  construct  a  section  having 
heights  that  are  means  between  corresponding  end  heights,  but 
for  very  irregular  sections  there  may  be  uncertainty  as  to  what 
heights  must  be  averaged  to  obtain  the  mid-section  heights.  For 
any  other  than  the  mid-sections  the  heights  are  obtained  with 
more  difficulty. 


CONSTRUCTION. 


195 


225.  Form  of  Notes.— The  record  of  ureas  find  volumes  may 
be  kept  in  the  form  below,  which  represents  the  cross-section 
book, with  the  necessary  columns  added. 


Sta. 

Ground 

Grade. 

L. 

C. 

R. 

End 
areas. 

Mid- 
areas. 

Exc. 

cu.yds. 

Emb. 
cu.yds. 

01 

1SS  r» 

ico  0 

+6.2 

1    fi     % 

+8.9 

+  1~5  53 

15.2 

17.9 

+4.1 

+5.3 

+7.5 

+130.64 

13.1 

16.5 

92 

185.4 

181.0 

4-2.0 
11.0 

+4.1 

+6.1 
15  1 

+89.96 

486.4 

•  1.0 
10.0 

+2.1 

+  3.0 
12.0 

+41.10 

93 

180.0 

180.0 

0.0 

0.0 

0.0 

0.0 

158.8 

9.0 

9.0 

-2  1 

-3.0 

-4.3 

-57.95 

10.2 

13.5 

94 

173.0 

179.0 

-4.2 
13.3 

-6.0 

-8.6 
19.9 

-144.40 

232.2 

If  the  method  of  averaging  end  areas  is  employed,  the  column 
of  mid-areas  will  not  be  needed,  and  may  even  be  omitted  when 
computing  by  the  prismoidal  formula.  In  this  case  the  notes  for 
mid-section  and  the  mid-area  should  be  written  in  red  ink. 

An  office  record  should  be  kept  in  addition  to  the  record  in  the 
cross-section  book,  to  which  it  will  not  be  necessary  to  transfer 
the  elevations  of  ground  and  grade.  If  preferred,  the  areas  and 
volumes  may  be  kept  ouiy  in  the  office  record,  omitting  them  in 
the  cross-section  book. 

226.  Prismoidal  Correction.—  The  time  and  labor  required  to 
obtain  the  area  of  the  mid-section  makes  the  use  of  the  prismoidal 
formula  objectionable;  for  this  reason  the  method  of  averaging 
end  areas  is  most  often  employed.  The  difference  in  the  two 
methods  will  not  be  great,  provided  the  difference  in  end  heights 
is  not  over  3  or  4  ft. ;  it  should  never  exceed  5  ft. 

When  the  difference  exceeds  this  a  considerable  error  is  intro- 
duced by  the  use  of  (326).  It  will  generally  be  sufficient  to 
average  end  areas  and  then  apply  a  correction  if  the  result  must 
be  free  from  large  errors. 

(a)  Correction  for  Level  Sections. — Between  two  level  end 
sections  the  volume  is  made  up  of  one  prism,  one  wedge,  and  two 
pyramids.  For  the  prism  and  wedge  the  true  volume  is  given  by 


196     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


averaging  end  areas,  but  for  the  pyramids  the  error   is  easily 
shown  to  be 

-7l0)8(l  l\ 

—>  —  IT    cubic  feet, 


or  <?=  C^-/*o)2  5^37  cubic  yards.     .     .     .     (329) 

The  table  below  gives  the  correction  C  in  cubic  yards,  com- 
puted by  (329),  for  a  few  values  of  H0  -  Aa ,  when  a  =  1  and  I  = 

100  ;  for  any  other  length  and  slope  multiply  by  ~^~ . 

100 

TABLE   OF   PRISMOIDAL    CORRECTIONS   FOR   LEVEL    SECTIONS. 


f 

H0-h0. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0 

0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

05 

1 

0 

G 

0.7 

0.9 

1.0 

1.2 

1.4 

1.6 

1.8 

20 

2.2 

2 

2 

5 

2  7 

3.0 

3.3 

3.6 

3,8 

4.2 

4.5 

4  8 

5.2 

3 

5 

6 

5'9 

6.3 

6.7 

7.1 

76 

8.0 

85 

8.9 

9.4 

4 

9 

9 

10.4 

10.9 

11.4 

12.0 

12.5 

13.1 

13.6 

14.2 

14.8 

5 

15 

4 

16.1 

16  7 

17.3 

18.0 

18.7 

10.4 

20.1 

20.8 

21.5 

6 

22 

.) 

23.0 

23.7 

24.5 

25.3 

26.1 

26.9 

27.7 

28.5 

29.4 

7 

30 

2 

31.1 

32.0 

32.9 

33.8 

34.7 

35.7 

36.6 

37.6 

38.5 

8 

39 

5 

40.5 

41.5 

42.5 

43.6 

44.6 

45.7 

46.7 

47.8 

48.9 

9 

50 

0 

51.1 

52.2 

58.4 

54.5 

55.7 

56.9 

58.1 

59.3 

60.5 

10 

61 

7 

63.0 

64.2 

65.5 

66.8 

68.1 

69.4 

70.7 

72.0 

73.3 

11 

74 

7' 

76.1 

•77.4 

78.8 

80.2 

81.6 

83.1 

84.  5 

86.0 

87  4 

12 

88 

9 

90.4 

91.9 

93.4 

94.9 

96.5 

98.0 

99.6 

101.1 

102  7 

13 

104 

3 

105.9 

107.6 

109.2 

110.8 

112.5 

114.2 

115  9 

117.6 

119.3. 

14 

121 

0 

122.7 

124.5 

126.2 

128.0 

123.8 

131.6 

133.4 

135.2 

137.0 

15 

138 

9 

140.7 

142.6 

144.5 

146.4 

148.3 

150.2 

152.2 

154.1 

156.1 

(b)  Correction  for  Three-level  Sections. — Formula  (329)  or 
the  foregoing  table  may  be  used  in  determining  the  correction 
for  three-level  sections  when  these  sections  are  somewhat  similar 
and  the  corresponding  heights  not  very  different.  A  general 
formula  may,  however,  be  derived. 

Let  the  center  and  side  heights  at  one  section  be  H0,  HI,  and 
HI,  respectively,  and  let  the  distance  between  slope-stakes  be 
W  =  Di  -{-  D-2\  let  the  corresponding  heights  at  the  other  end 
section  be  7ia,  7tlf  and  7i2  with  a  distance  between  slope-stakes  of 
w  =  tf,  -f-  d-2. 

By  formula  (321)  the  areas  will  be: 

Area  at  first  end  F  ?(//,  +  //,)  +  r?4i»  •    •    «     W 


CONSTRUCTION.  197 


Area  at  second  end  =  ^(7^,  -f  7<.a)  -j-  T7/o,     .     .     .(b) 


4  X  mid.area  =  4L 


. 
From  (a),  (5),  and  (c)  the  volume  by  the  prismoidal  rule  is 

V  =  [f  &(//!  +  Ai  +179+/*a)+  F  (^0+^" 
From  (a)  and  (6),  by  averaging  end  areas, 

'  + 

TTfi;  +  |w0    X     . 
From  (d  )  and  (e)  the  correction  is 

F'  -  V  =    -?(//  -  /    +    (A 


=  (H0  -  7t0)(W  -  w)  x       cu.  ft.  .     ,     (330) 
The  correction  in  cubic  yards  is 

C  =  (H0  -  hQ)(W-  w}  X  12^;    •     .     (331) 

when  1=  100, 

C  =  Q.3l(II0  -  h0)(W  -  w).     .     .     .     .     (331') 

This  correction  may  be  either  positive  or  negative. 
EXAMPLE.  —  Compute  the  correction  for  the   two    prismoids 
below. 

Sta.  L.  c.  R. 

160..  ±-4        +3.0 

161 

162  ...... 


15.6 
+10'8  +11'2 


_ 
14.2  13.8 


198     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


For  the  prismoid  between  160  and  161 


C=  (3.0  -  7.0)(30.0  -  40.0)  X 


100 


13  X  27 


-  =  +12.3cu.  yds. 


For  the  prismoid  between  161  and  163 

100 
C  =  (7.0  -  9.0)(40.0  -  28.0)  X  19  v  >>7  =  -  7.4  cu.  yds. 

Aw     /\     *J  i 

When  the  correction  C  is  positive  it  must  be  subtracted  from, 
and  when  negative  added  to,  the  volume  as  found  by  averaging 
end  areas  ;  this  is  evident  from  (330).  6' will  be  negative  when 
the  smaller  center  height  and  greater  width  between  slope-stakes 
occur  at  the  same  station,  as  is  illustrated  in  the  example. 

The  general  tendency,  however,  with  the  method  of  averaging 
end  areas  is  to  give  volumes  that  are  too  large,  and  the  error  in- 
creases with  the  square  of  the  difference  in  end  heights,  as  is  evi- 
dent from  (329). 

227.  When  the  center  and  sides  of  roadbed  do  not  pass  from 
cut  to  fill  at  the  same  station  we  have  a  volume,  part  excavation, 
part  embankment,  between  the  same  pair  of  sections,  such  as  is 
illustrated  in  Fig.  106. 

Between  sections  ABC  and  EDFML  the  solid  having  bases 


FIG. 106. 

ABC  and  FED  is  embankment,  while  the  pyramid  FML-A  is 
in  excavation. 

For  such  cases  as  this  the  method  of  averaging  end  areas  is 
most  in  error,  particularly  if  the  ground  have  a  sharp  longitudinal 


CONSTRUCTION. 


199 


as  well  as  transverse  slope      Whatever  method  is  employed,  the 
excavatious  and  embankments  must  be  separately  computed. 

228.  Tables  of  Volumes  for  Level  Sections  and  Equal  End 
Areas  may  be  used  in  making  preliminary  estimates.     The  aver- 
age center  height  for  one  or  more  stations  is  taken  from  the  pro- 
file and  the  volume  at  once  read  off  from  tables,  such  as  Table 
XIX. 

Table  XX  may  be  used  in  finding  the  volume,  after  having 
averaged  the  end  areas,  and  a  correction  made  by  226  if  desired. 

229.  Side  Ditches  in  cuts  have  a  constant  cross-section,  and 
hence  a  constant  volume  for  each  full  station.     Their  contents 
are  separately  computed  and  added  after  the  other  computations 
have  been  made.    They  need  not  be  shown  in  cross-section  notes. 

230.  Earthwork   on   Curves. — In    computing  quantities   on 
curves  the  end  sections  are  assumed  to  be  parallel,  and  the  axial 
distance  between  sections  taken  as  the  length  of  the  prismoid. 
If  the  volume  be  taken  as  generated  by  a  moving  section,  and  the 
center  of  gravity  of  this  section  lie  always  on  a  vertical  Hue  passing 
through  the  axis,  this  method  gives  correct  results  ;  otherwise  not. 
The  result  will  be  too  small  or  too  large  according  as  the  center  of 
gravity  falls  without  or  within  the  center  line  of  curve. 

If  the  volumes  are  computed  by  averaging  end  areas,  it  will  be 
a  useless  refinement  to  apply  a  curvature  correction  ;  but  if  the 
prismoidal  formula  is  employed,  and  accuracy  is  desired,  it 
should  be  applied,  especially  if  the  work  be  in  rock. 


FIG.  107. 

To  find  the  curvature  correction  (c.c.)  consider  Fig.  107,  which 
represents  the  mean  section  of  the  prismoid. 


200     A  FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 

The  portion  ABHEG  has  its  center  of  gravity  on  the  line  BF 
(BII  having  the  same  slope  as  BA) ;  hence  the  path  of  its  center 
of  gravity  will  be  the  same  length  as  the  axis  of  the  prismoid, 
and  there  will  be  no  error  in  the  computed  volume  generated  by 
this  portion.  In  the  triangle  BCHdvuw  BK  to  the  mid-point  of 
CH.  The  center  of  gravity  of  this  triangle  is  at  M,  two  thirds  of 
the  distance  BK  from  B.  Now,  by  Guldiu's  rule  (theorem  of 
Pappus)  the  volume  generated  equals  the  area  multiplied  by  the 
path  of  the  center  of  gravity,  the  center  of  rotation  being  in  the 
plane  of  the  area. 

Draw  BL  horizontal  and  take  N  on  a  vertical  through  M ;  let 
the  angle  in  degrees  at  the  center  be  6. 

The  volume  generated  by  the  triangle  BCH  is 


But  the  calculated  volume  is 

f  ° 

180 
Hence  the  curvature  correction  will  be 


. -.  c.c.  =  --BCI^d,  f  d,)0°  =  .QOQBCn(dl  -f  d,)Q°. 
o40 

When  the  sections  are  100  ft.  apart  6°  =  D  and  the  correction 
becomes 

c.c.  =  .0065  CIl(di+d*)D (302') 

The  area  of  the  triangle  JRCHis  easily  seen  to  be 

A  =  t[b(ha  -  A,)  +  7t0(da  -  d,)].     .     .     (333) 

If  the  triangle  BCII  is  on  the  convex  side  of  curve  the  correc- 
tion must  be  added,  if  ou  the  concave  side  it  must  be  subtracted. 

For  light  work  the  correction  is  small,  but  for  heavy  work  with 
steep  transverse  slope  on  sharp  curves  it  may  be  considerable. 
In  practice  we  may  use  the  mi 1  for  the  mean  area  without 
material  error. 

EXAMPLE.— Find  the  correction  per  station  on  an  8°  curve,  28 


CONSTRUCTION 


201 


ft.  base,  side  slopes  1|  to  1,  inside  height  10  ft.,  outside  height 
30  ft.,  end  sections  equal. 

231.  Overhaul. — Contract  prices  are  usually  based  on  a  certain 
maximum  length  of  haul,  and  all  material  carried  farther  than 
this  is  termed  overhaul,  for  which  the  contractor  receives  extra 
compensation. 


FIG.  108 

In  Fig.  108  let  AB  be  the  length  of  free  haul,  the  points  A  and 
B  being  fixed  on  the  profile  so  that  the  volume  ACE  equals  the 
volume  CBK;  this  may  be  done  by  trial  computations,  or  closely 
enough  in  some  cases  from  the  profile  alone.  Let  the  mass 
EFHA  be  removed  to  BKLM,  and  let  the  centers  of  mass  in  the 
two  positions  be  at  g  and  ffi  respectively;  the  length  of  overhaul 
to  be  paid  for  will  be  GG±  -  AB  =  GA  +  BG,.  To  find  g  and 
#!  accurately  requires  that  the  sum  of  the  moments  of  the  ele- 
mentary masses  equal  the  moment  of  the  whole  mass  with  respect 
to  any  chosen  point.  It  will  answer  in  practice  to  multiply  the 
volume  per  station  by  the  distance  of  its  center  of  mass  (found 
by  dividing  the  station  length  in  the  inverse  ratio  of  its  end  areas) 
from  some  selected  point,  as  C,  and  equate  this  to  the  product  of 
the  whole  mass  by  the  unknown  distance  of  its  center  of  gravity 
from  the  same  point,  then  solve  for  this  distance.  Indeed,  it  will 
answer  in  most  cases  to  find  a  point  that  divides  the  mass  into 
two  equal  parts  and  treat  this  as  the  center  of  gravity;  such  a 
point  may  be  readily  found  by  trial. 

Sometimes  it  is  specified  that  the  overhaul  must  be  found  by 
finding  the  distance  of  the  center  of  gravity  of  the  whole  mass 
moved  from  the  center  of  gravity  of  the  same  mass  after  deposit- 
ing in  embankment  and  deducting  from  this  the  length  of  free 
haul,  the  remainder  being  called  the  overhaul.  This  is  the  easier 
method,  but  requires  every  yard  moved  to  be  carried  the  entire 
length  of  free  haul  before  any  overhaul  whatever  is  counted. 

EXAMPLE.— Let  AEFH  =  5000  cu.  yds.,  GA  =  200  ft.,  G,B 


202     A   FIELD-MAKUAL  FOR   HAILROAD   ENGINEERS. 

=  300  ft.,  and  the  price  paid  for  overhaul  1  cent  per  cubic  yard 
per  100  ft. 

The  additional  compensation  above  the  contract  price  will  be 

20°1|0800  X  5000  X  .01  =  $250.00. 


ARTICLE  19.  GRADE  AND  BALLAST  STAKES,  CULVERTS, 
BRIDGES,  AND  TUNNELS. 

232.  Grade  and  Center  Stakes.— After  the  excavations  and 
embankments  have   been   brought   approximately   to  the   level 
called   for  on   the   cross-section  stakes,    the   engineer  must  go 
carefully  over  the  road,  setting  center  stakes  every  hundred  feet 
on  tangents  and  flat  curves,  and  every  5C  or  25  feet  on  sharp 
curves— the  distance  between  center  stakes  depending  on  the 
sharpness  of  the  curves.     On  tangents  it  will  be  sufficient  to  drive 
a  grade-stake  beside  each  center  stake,  so  that  its  top  will  be  at 
the  height  to  which  the  finished  surface  must  come,  due  allow- 
ance being  made  for  shrinkage. 

On  curves  grade-stakes  must  be  set  at  each  side  a  distance 
equal  to  the  half-base  from  the  center;  the  proper  elevation  or 
depression  of  these  stakes  must  be  found  by  207,  formula  (320). 

The  P.C.'s  and  P.T.'s  are  recovered  by  means  of  the  reference- 
points  set  during  location. 

233.  Ballast-stakes  are  set  on  the  completed  sub-grade  at  the 
proper  width  of  ballast-base — just  as  in  slope-staking — with  their 
tops  at  the  level  of  the  final  grade.     They  should  be  set  at  inter- 
vals of  50  ft.  on  tangents  and  flat  curves,  and  at  25  ft.  on  sharp 
curves. 

234.  Track  Centers  are  set  for  the  guidance  of  trackmen  as 
soon  as  the  road-bed  is  ready  to  receive  the  cross-ties  and  rails. 

235.  The  Opening  left  for  a  culvert,  drain,  or  trestle  bridge  is 
measured  from  top  of  bank  to  top  of  bank;  the  manner  in  which 
it  should  be  staked  out  is  described  in  213. 

A  note  of  the  size  of  drain  and  the  material  of  which  it  is  to  be 
built,  whether  glazed  earthenware  pipe,  box  drain,  stone  culvert, 
etc.,  should  be  made  in  the  note-book  opposite  the  notes  for  the 
opening. 

After  the  culvert  or  drain  has  been  built  the  earth  is  filled  in 


CONSTRUCTION.  203 

over  and  around  it,  and  face  or  wing  walls  built  to  protect  the 
bank  at  the  points  where  culvert  or  drain  meets  its  face. 

For  trestle  bridges  it  must  be  remembered  that  the  bank-sills 
set  back  from  the  top  of  bank  a  distance  sufficient  to  give  firm 
bearing,  usually  about  6  ft.  for  ordinary  earth,  and  allowance 
made  therefor  in  staking  out  the  opening.  The  length  of  open- 
ing is  designated  by  the  number  of  bents  between  bank-sills: 
thus  a  12-bent  opening,  where  the  distance  between  bents  is  14 
ft.,  would  be  13  X  14  -  12  =  170  ft.  The  bent  spacing  depends 
upon  the  size  of  timbers  available  and  upon  the  weight  of  loco- 
motives to  be  run  over  the  road. 

Whatever  the  nature  of  the  structure,  ample  waterway  should 
always  be  provided  for  the  heaviest  storms;  failure  to  do  this  is 
the  cause  of  many  a  costly  wreck. 

Center  stakes  are  set  for  each  trestle-bent,  and  if  piles  are  to  be 
driven  a  stake  should  mark  the  position  of  each  pile.  If  the 
bridge  is  not  at  right  angles  to  the  stream  it  will  often  be  best  to 
set  the  bents  askew,  but  this  should  be  avoided  whenever  possible. 
After  the  piles  have  been  driven  cut-off  levels  are  given  by  the 
engineer,  for  which  a  tack  is  set  in  the  pile  at  a  definite  distance 
below  the  point  of  cut-off,  allowance  being  made  for  cap, 
stringer,  etc.  If  the  bridge  is  on  a  grade,  the  rate  of  rise  per  bent 
must  be  figured  out  -and  allowed  for.  On  curves  the  proper 
superelevation  of  outer  rail  must  be  computed  by  the  method  of 
207. 

For  details  of  trestles  see  Foster's  Trestle  Bridges. 

236.  The  Piers  and  Abutments  for  truss  bridges  must  be  very 
accurately  located,  the  spacing  being  done  with  a  steel  tape 
whose  constants  are  known,  and  the  center  and  limits  being 
marked  by  stakes.  On  tangents  the  centers  are  easily  located 
and  referenced,  but  on  curves  this  is  not  so  easy,  as  the  center  of 
track  cannot  be  taken  as  the  center  of  pier  on  account  of  the 
clearance  necessary  for  trains. 

Bridges  on  curves  should  be  avoided  whenever  possible,  but 
when  they  cannot  be  avoided  the  centers  of  piers  are  to  be  placed 
at  the  intersection  of  pier-axis  and  "bridge-chord." 

In  Fig.  109  ABC  is  the  center  line  of  track,  AE  and  CF  the 
pier-axes.  At  the  mid-point  of  the  arc  AC  the  tangent  EF, 
parallel  to  AC,  is  drawn;  make  AN '  =  NE  =  CL  =  LF,  and  draw 
NL,  which  is  the  bridge-chord.  The  points  N  and  L  are  the 
centers  of  the  piers. 


204     A   FIELD-MANUAL  FOR  RAILROAD   ENGINEERS. 

Should  L  or  N  be  inaccessible,  they  may  be  located  from  a 
point  P  on  some  accessible  portion  of  the  curve.  To  do  this  take 
PQ  perpendicular  to  LN,  such  that 


-ls  versa);   .     .     (334) 
then  will 

QL  =  QK+  KL  =  £(sin  &  +  sin  a).     ...     (335) 

The  manner  of  building  the  piers,  determining  the  nature  of 
the  foundation,  and  erecting  the  bridge  come  properly  within 


the  province  of  the  bridge  engineer  and  require  too  much  space 
to  be  outlined  here.  For  preliminary  estimates  it  will  often  be 
sufficient  for  the  locating  engineer  to  make  soundings  with  gas- 
pipe  in  order  to  determine  the  depth  to  a  suitable  foundation 
and  the  nature  of  the  overlying  deposits,  the  core  forced  up 
within  the  pipe  serving  for  the  latter  purpose. 

237.  Tunnels,  like  bridges,  require  great  nicety  in  the  meas- 
urements by  which  they  are  constructed.  The  angular  measure- 
ments should  be  made  with  the  best  available  transit  in  the  best 
possible  adjustment,  and  repetitions  and  reversals  made  to  elimi- 
nate errors  as  much  as  possible.  Linear  measurements  should  be 
made  with  a  steel  tape  the  constants  of  which  are  known,  so  that 
correction  may  be  made  for  temperature,  etc. 

If  headings  are  to  be  driven  from  the  ends  and  an  unobstructed 
view  of  the  summit  is  obtainable,  a  point  may  be  fixed  in  the 
same  vertical  plane  as  the  axis  of  the  road,  and  will  serve  for 
giving  the  alignments  of  the  headings. 

Sometimes  several  points  must  be  located  on  the  mountain  in 
the  plane  of  the  axis,  and  triangulation  resorted  to  to  secure  the 
desired  end. 


CONSTRUCTION.  205 

The  most  accurate  work  can  often  be  done  at  night,  sightings 
being  made  to  a  plummet-lamp,  or  in  the  early  morning  before 
the  sun's  heat  has  produced  great  changes  iu  the  density  of  the 
air. 

Within  the  tunnel,  alignment  is  made  by  sighting  to  a  plummet- 
lamp  suspended  from  a  plug  M  into  the  roof.  Work  is  usually 
carried  on  from  both  ends,  so  that  it  is  necessary  to  secure  accurate 
alignment.  If  the  entire  tunnel  is  on  a  tangent,  this  is  not 
difficult  when  working  from  the  ends;  but  when  the  tunnel  is  on 
a  curve  (the  curve  falling  most  often  at  the  ends),  or  when  align- 
ment must  be  transferred  down  a  shaft,  the  operation  is  much 
more  difficult. 

The  Mont  Ceuis  Tunnel,  over  seven  miles  long,  was  constructed 
from  the  ends — one  end  being  on  a  curve — yet  there  was  no 
trouble  in  making  a  fit  where  the  headings  met. 

When  headings  are  driven  from  the  foot  of  a  shaft  it  is 
necessary  to  secure  a  point  in  the  surface  on  each  side  of  shaft 
in  the  plane  of  tunnel-axis  and  to  transfer  these  points  by  plumb- 
lines  to  the  bottom.  By  connecting  these  transferred  points  the 
direction  of  the  axis  may  be  secured.  In  the  Hoosac  Tunnel  a 
line  was  transferred  down  a  shaft  1000  ft.  deep  and  carried  2050 
ft.  with  a  final  error  of  only  nine  sixteenths  of  an  inch. 

Levels  are  run  over  the  surface  with  great  care,  and  may  be 
transferred  down  a  shaft  by  measuring  its  depth  with  a  rod  or 
steel  tape.  The  grade  of  the  bottom  must  be  sufficient  for 
drainage. 

The  dimensions  of  the  tunnel  will  depend  on  the  height  of 
engines  and  the  purposes  for  which  it  was  intended. 

For  detailed  information  regarding  tunnels  the  student  is 
referred  to  Drinker's  and  Sims's  books  on  the  subject  and  to  the 
current  engineering  journals. 

ARTICLE  20.  MONTHLY  AND  FINAL  ESTIMATES. 

238.  Monthly  Estimates  are  made  by  the  engineers  in  charge 
of  construction  about  the  end  of  each  month,  and  upon  these  the 
division  engineer  bases  his  estimate,  which  he  forwards  to  the 
chief  engineer  for  approval.  The  contractor  receives  his  compen- 
sation some  days  later,  usually  about  the  15th  or  20th  of  the 
month  following.  Monthly  estimates  should  always  be  based  on 
actual  measurements  and  never  guessed  at,  particularly  if  several 
classifications  arc  to  be  made,  The  total  quantity  of  work  done 


206     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

or  material  delivered  is  to  be  estimated,  then  the  difference 
between  any  estimate  and  the  last  preceding  one  will  be  the 
estimate  upon  which  the  contractor  receives  his  installments. 

239.  For  Earthwork,  measurements  (when  needed)  are  only 
approximate,  but  it  is  best  to  make  them  with  level  and  tape 
even  for  monthly  estimates.     It  will  be  sufficient  to  compute 
volumes  by  averaging  end  areas,  no  attention  being  paid  to  the 
prismoidal  correction.     Care  must  be  taken,  however,  that  such 
estimates  are  not  in  excess — in  fact,  it  is  well  to  keep  slightly 
within  the  actual  quantities  on  account  of  the  greater  cost  and  labor 
required  to  finish  the  work,  which  would  make  the  latter  part 
appear  so  much  less  profitable  to  the  contractor  as  sometimes  to 
induce  a  disposition  to  abandon  the  work  before  completion. 

240.  The  Classification  of  Earthwork. — It  is   customary  to 
group  earthwork  in  excavation,  according  to  the  difficulty  of 
removal,   into   three   classes— earth-excavation,  loose   rock,    and 
solid  rock— though  other  classifications  are  frequently  made. 

Earth-excavation  includes  all  earth,  sand,  loani,  and  loose 
stones  that  can  be  moved  with  the  plow  and  scraper. 

Loose  rock  includes  all  stones  and  detached  boulders  less  than 
from  1  to  3  cubic  yards  in  size,  and  all  slate,  shale,  or  cemented 
gravel  requiring  the  use  of  the  pick  and  bar,  but  which  may  be 
removed  without  blasting. 

Solid  rock  includes  all  boulders  above  a  certain  size  (usually 
from  1  to  3  cubic  yards,  as  specified)  and  all  rock  masses  that 
cannot  be  removed  without  blasting. 

The  relative  prices  vary,  but  a  ratio  of  1 :  3  :  7  will  not  be  far 
from  an  average  for  the  more  common  conditions  arising  in  rail- 
road work. 

The  necessity  for  a  correct  classification  is  evident,  and  the 
engineer  should  keep  full  notes,  and  make  careful  measurements 
whenever  a  given  volume  involves  more  than  one  class  of  earth- 
work. It  is  customary  to  specify  that  his  decision  is  final,  and 
therefore  his  measurements  should  be  carefully  made  during  the 
progress  of  the  work,  and  notes  on  the  nature  of  material  taken 
at  the  same  time. 

A  note-book  should  be  kept  showing  the  measurements  and 
amounts  of  each  class  of  material  for  each  station,  together  with 
the  date  of  completion  and  acceptance. 


*:  1ITYJ    2W 


241.  A  Progress  Profile  should  accompTm^ffitfmouthly  esti- 
mate to  exhibit  graphically  the  amount  of  work  done  during  the 
mouth,  different  colors  being  used  for  the  different  months.     The 
final  profile  should  show  approximately  the  progress  of  the  work. 
The  colors  may  be  laid  on  with  a  brush,  or  hatchings  made  with  a 
pen;  in  neither  case  should  the  color  obscure  the  lines  of  the  pro- 
file-paper.    A  duplicate  progress  profile  should  be  retained  in  the 
division  engineer's  office;  if  transparent  profile-paper  is  employed, 
one  may  be  simply  traced  through  from  the  other.     A  further 
advantage  of  the  transparent  paper  is  that  blue-prints  of  any  por- 
tion of  the  profile  may  be   readily  made  wneii  duplicates  are 
desired,  provided  the  drawings  are  in  black  or  any  color  admit- 
ting blue  printing. 

242.  Masonry  is  to  be   measured  in   cubic  yards,  and  any 
material  on  hand,  but  not  in  place,  is  to  be  measured  and  esti- 
mated.    The   classification    of   masonry   must   be   according  to 
specifications.     Foundation-pits  for  piers    or    culverts   must  be 
measured  as  soon  as  completed,  and  before  the  masonry  has  been 
put  in  place. 

243.  Bridges    must    be    estimated    by    measurement,    or   by 
checking  up  material  in  place  and  that  on  hand  but  not  in  place. 

For  trestle  bridges,  or  foundations  requiring  piling,  the  actual 
number  of  linear  feet  below  cap  must  be  measured;  this  neces- 
sitates the  constant  supervision  of  the  engineer  or  an  assistant, 
sometimes  known  as  a  "pile-recorder,"  whose  duty  it  is  to  see 
that  all  piles  come  up  to  specifications  and  are  driven  in  accord- 
ance therewith. 

All  framing-timber  in  place,  or  delivered  but  not  in  place,  is 
to  be  included  in  the  estimate,  the  amount  being  obtained  by 
measurement. 

Steel  spans  or  trestles  are  to  be  estimated,  in  the  same  manner 
as  wooden  trestles,  by  checking  up  or  measuring  the  material  on 
hand  and  in  place. 

244.  Track  Material  must  be  checked  up  either  by  the  '  '  mate- 
rial clerk  "  or  the  engineer  in  charge  of  track.     Ballasting  prop- 
erly belongs  with  the  graduation,  but  may  be  put  in  place  after 
the  rails  have  been  laid;  in  either  case  it  is  estimated  in  accord- 
ance with  the  specifications. 

For  preliminary  and  monthly  estimates  it  \vill  be  sufficient  to 


208     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

estimate  track  material  by  menus  of  tables  showing  the  number 
of  cross-tics  for  a  given  spacing  and  the  weight  of  steel  for  a 
given  rail  section,  but  before  the  final  estimate  is  made  all  mate- 
rial must  be  measured  or  counted. 

245.  Blank  Estimate-sheets  are  sent  out  from  the  chief  engi- 
neer's office  to  be  filled  out  by  the  engineers  making  estimate, 
who  should  retain  a  copy  of  each  estimate  rendered.     On  these 
sheets  should  appear  the  total  quantity  estimated,  the  amount  of 
thelast  preceding  estimate,  and  the  estimate  for  the  month,  which 
will  be  the  difference  of  the  other  two. 

The  division  engineer's  estimate  must  show  not  only  the  quan- 
tity of  material,  but  its  value  in  dollars  and  cents  computed  from 
the  contract  price.  The  footings  of  the  several  columns  then 
serve  as  a  check  upon  each  other. 

246.  The   Monthly   Payments    are   not    made   for  the   full 
amount  estimated,  but  about  15  or  20  per  cent  is  retained  until 
after  the  final  estimate  has  been  made,  in   order  to  insure  the 
completion  of  the  work  by  the  contractor,  and  to  be  used  as  a 
fund  from  which  to  withhold  the  amount  of  damages  provided  in 
the  contract  for  failure  to  comply  with  all  its  provisions. 

247.  Extras  incident  to  minor  changes,  or  to  the  protection  or 
drainage  of  the  work,  are  usually  shown  on  the  final  estimate,  but 
a  better  way  would  be  to  require  the  contractor  to  present  his  bill 
for  extras  at  the  end  of  each  month,  and  to  incorporate  them  in 
the  monthly  estimate  when  they  are  just.     The  engineer  should 
take  measurements  upon  any  extra  work  at  the  time  of  its  com- 
pletion, and  should  keep  a  record  thereof.     If  the  extras  are  of  a 
nature  not  admitting  of  measurement,  he  should  note  the  com- 
pensation to  be  allowed  at  the  time  the  extra  work  is  done. 

248.  The  Final  Estimate  must  include  all  earthwork  moved, 
all  material  in  bridges,  all  masonry  in  foundations,  culverts,  piers, 
and  tunnels,  and  all  other  material  supplied   or  work  done  in 
compliance  with  the  contract.     The   engineer  should  keep  his 
notes  full  and  complete  during  the  construction  of  the  work,  in 
order  to  be  able  to  meet  the  contractor's  claims  for  extras  or  com- 
plaints as  to  classification.     Any  items  that  may  have  been  over- 
looked in  making  up  the  monthly  estimates  must  be  included 
here. 


CONSTRUCTION.  209 

249.  Acceptance. — Until  the  engineer  has  pronounced  the 
work  satisfactory  and  formally  accepted  it  the  contractor  is 
liable  for  its  condition,  and  must  make  good  all  damage  caused 
by  accident  or  storm.  The  road-bed  and  track  may  be  accepted 
without  special  test;  but  all  spans  should  be  subjected  to  a  speci- 
fied test-load,  under  which  they  must  show  not  more  than  a  cer- 
tain maximum  deflection,  so  their  acceptance  will  come  last. 

Sometimes  the  contract  requires  a  particular  structure  or  class 
of  structures  to  be  maintained  in  good  order  for  a  certain  length 
of  time  after  completion,  and  a  percentage  is  retained  to  cover 
the  case. 

After  final  acceptance  the  work  is  paid  for  in  accordance  with, 
the  final  estimate. 


TABLES. 


212 


TABLE  I.— RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

0°  0' 

Infinite  ' 

1°  0' 

5729.65 

2°  0' 

2864.93 

3°  0' 

1910.08 

4°  0' 

1432.69 

i 

343775. 

1 

5635.72 

1 

2841.26 

l 

1899.53 

1 

1426.74 

2 

171887. 

2 

5544.83 

2 

2817.97 

2 

1889.09 

2 

1420.85 

3 

114592. 

3 

5456.82 

3 

2795.06 

3 

1878.77 

3 

1415.01 

4 

85943.7 

4 

5371.56 

4 

2772.53 

4 

1868.56 

4 

1409.21 

5 

68754.9 

5 

5288.92 

5 

2750.35 

5 

1858.47 

5 

1403.46 

6 

57295.8 

6 

5208.79 

6 

2728.52 

6 

1848.48 

6 

1397.76 

49110.7 

7 

5131.05 

7 

2707.04 

7 

1838.59 

7 

1392.10 

8 

42971.81 

8 

5055.59 

8 

2685.89 

8 

1828.82 

8 

1386.49 

9 

38197.2' 

9 

4982.33 

9 

2665.08 

9 

1819.14 

9 

1380.92 

10 

34377.  5  ' 

10 

4911.15 

10 

2644.58 

10 

1809.57 

10 

1375.40 

11 

31252.3 

11 

4841.98 

11 

2624.39 

11 

1800.10 

11 

1369.92 

IS 

28647.8 

12 

4774.74 

12 

2604.51 

12 

1790.73 

12 

1364.49 

13 

26444.2 

13 

4709.33 

18 

2584.93 

13 

1781.45 

13 

1459.10 

14 

24555.4 

14 

4645.69 

14 

2565.65 

14 

1772.27 

14 

1353.75 

15 

22918.3 

15 

4583.75 

15 

2546.64 

15 

1763.18 

15 

1348.45 

16 

21485.9 

16 

4523.44 

16 

2527.92 

16 

1754.19 

16 

1343.15 

17 

20222.1 

17 

4464.70 

17 

2509.47 

17 

1745.26 

17 

1337.65 

18 

19098.6 

18 

4407.46 

18 

2491.29 

18 

1736.48 

18 

133-2.77 

19 

18093.4 

19 

4351.67 

19 

2473.37 

19 

1727.75 

19 

1327.63 

20 

17188.8 

20 

4297.28 

20 

2455.70 

20 

1719.12 

20 

1322.53 

21 

16370.2 

21 

4244.23 

21 

2438.29 

21 

1710.56 

21 

1317.46 

22 

15626.1 

22 

4192.47 

22 

2421.12 

22 

1702.10 

22 

1312.43 

23 

14946.7 

j   23 

4141.96 

23 

2404.19 

23 

1693.72 

23 

1307.45 

24 

14323.6 

24 

4092.66 

24 

2387.50 

24 

1685.42 

24 

1302.50 

25 

13751.0 

25 

4044.51 

25 

2371.04 

25 

1677.20 

25 

1297.58 

26 

13222.1 

26 

3997.49 

26 

2354.80 

26 

1669.06 

26 

1292.71 

27 

12732.4 

27 

3951.54 

27 

2338.78 

27 

1661.00 

27 

1287.87 

28 

12277.7 

28 

3906.54 

28 

2322.98 

28 

1653.01 

28 

1283.07 

29 

11854.3 

29 

3862.74 

29 

2307.39 

29 

1645.11 

29 

1278.30 

30 

11459.2 

30 

3819.83 

30 

2292.01 

30 

1637.28 

30 

1273.57 

31 

11089.6 

31 

3777.85 

31 

2276.84 

31 

1629.52 

31 

1268.87 

32 

10743  0 

32 

3736.79 

32 

2261.86 

32 

1621.84 

32 

1264.21 

33 

10417.5 

33 

3696.61 

33 

2247.08 

33 

1614.22 

33 

1259.5S 

34 

10111.1 

34 

3657.29 

34 

2232.49 

34 

1606.68 

34 

1254.98 

35 

9822.18 

&5 

3618.80 

35  2218.09 

35 

1599.21 

35 

1250.42 

36 

9549.34 

36 

3581.10 

36 

2203.87 

36 

1591.81 

36 

1245.89 

37 

9291.29!   37 

3544.19 

37 

2189.84 

37 

1584.48 

37 

1241.40 

88 

9046.75 

38 

3508.02 

38 

2175.98 

38 

1577.21 

38 

1236.94 

39 

8814.78 

39 

3472.59 

39 

2162.30 

39 

1570.01 

39 

1282.51 

40 

8594.42 

40 

3437.87 

40 

2148.79 

40 

1562.88* 

40 

1228.11 

41 

8384.80 

41 

3403.83 

41 

2135.44 

41 

1555.81 

41 

1223.74 

42 

8185.16 

42 

3370.46 

42 

2122.26 

42 

1548.80 

42 

1219.40 

43 

7994.81 

43 

3337.74 

43 

2109.24 

43 

1541.86 

43 

1215.30 

44 

7813.11 

44 

&305.65 

44 

2096.39 

44 

1534.98 

44 

1210.82 

45 

7639.49 

45 

3274.17 

45 

2083.68 

45 

1528.16 

45 

1206.57 

46 

7473.42 

46 

3243.29 

46 

2071.13 

46 

1521.40 

46 

1202.36 

47 

7314.41 

47 

3212.98 

47 

2058.73 

47 

1514.70 

47 

1198.17 

48 

7162.03 

48 

3183.23 

48 

2046.48 

48 

1508.06 

48 

1194.01 

49 

7015.87 

49 

3154.03 

49 

2034.37 

49 

1501.48 

49 

1189.88 

50 

6875.55 

50 

3125.36 

50 

2022.41 

50 

1494.95 

50 

1185.78 

51 

6740.74 

51 

3097.20 

51 

2010.59 

'51 

1488.48 

51 

1181.71 

52 

6611.12 

52 

3069.55 

52 

1998.90 

52 

1482.07 

52 

1177.60 

53 

6486.38 

53 

3042.39 

53 

1987.35 

53 

1475.71 

53 

1173.65 

54 

6366.26 

54 

3015.71 

54 

1975.93 

54 

1469.41 

54 

1169.66 

55 

6250.51 

55 

2989.48 

55 

1964.64 

55 

1463.10 

55 

11(55.70 

56 

6138.90 

56 

2963.71 

56 

1953.48 

56 

1456.96 

50 

1161.76 

57 

603i.20 

57 

2938.39 

57 

1942.44 

57 

1450.81 

57 

1157.85 

58 

5927.22 

58  2913.49 

58 

1931.53 

58 

1444.72 

58 

1153.97 

59 

5826.76 

59 

2889.01 

59 

1920.75 

59 

1438.68 

59 

1150.11 

60 

5729.65 

60 

2864.  9o 

60  1910.08 

60 

1432.09 

60 

1146.28 

TABLES. 


213 


TABLE   I.— RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

6°0/ 

1146.28 

6°0' 

955.37 

70  0/ 

818.64 

8°0' 

716.34 

9°0' 

636.78 

1 

1142.47 

1 

952.72 

1 

816.70 

1 

714.85 

1 

635.61 

2 

1138.69, 

2 

950.09 

2 

814.76 

2 

713.37 

2 

634.44 

3 

1134.94' 

3 

947.48 

3 

812.83 

3 

711.90 

3 

633.27 

4 

1131.21 

4 

944.88 

4 

810.92 

4 

710.43 

4 

632.10 

5 

1127.50 

5 

942.29 

5 

809.01 

5 

708.96 

5 

630.94 

6 

1123.82 

6 

939.72 

6 

807.11 

6 

707.51 

6 

629.79 

7 

1120.16 

7 

937.16 

805.22 

7 

706.05 

7 

628.64 

8 

1116.52 

8 

934.62 

8 

803.34 

8 

704.60 

8 

627.49 

9 

1112.  91j 

9 

932.09 

9 

801.47 

9 

703.16 

9 

626.35 

10 

1109.33J 

10 

929.57 

10 

799.61 

10 

701.73 

10 

625.21 

11 

1105.76 

11 

927.07 

11 

797.75 

11 

700.30 

11 

624.08 

12 

1102.22 

12 

924.58 

12 

795.91 

12 

698.88 

12 

622.95 

13 

1098.70 

13 

922.10 

13 

794.07 

13 

697.46 

13 

621.82 

14 

1095.20 

14 

919.64 

14 

792.24 

14 

696.05 

14 

620.70 

15 

1091.73 

15 

917.19 

15 

790.42 

15 

694.65 

15 

619.58 

16 

1088.  28  1 

16 

914.75 

16 

788.61 

16 

693.24 

16 

618.47 

17 

1084.85 

17 

912.33 

17 

786.80 

17 

691.85 

17 

617.36 

18 

1081.  44  ! 

18 

909.92 

18 

785.01 

18 

690.46 

18 

616.25 

19 

1078.05 

19 

907.52 

19 

783.22 

19 

689.08 

19 

615.15 

20 

1074.68 

20 

905.13 

20 

781.44 

20 

687.70 

20 

614.05 

21 

1071.34 

21 

902.76 

21 

779.67 

21 

686.33 

21 

612.96 

22 

1068.  Ol1 

22 

900.40 

22 

777.91 

22 

684.96 

22 

611.87 

23 

1064.71 

23 

898.05 

23 

776.15 

23 

683.60 

23 

610.78 

24 

1061.43 

24 

895.71 

24 

77'4.40 

24 

682.25 

24 

609.70 

25 

1058.16 

25 

893.39 

25 

772.66 

25 

680.89 

25 

608.62 

26 

1054.92 

26 

891.08 

26 

770.93 

26 

679.55 

26 

607.55 

27 

1051.70 

27 

888.78 

27 

769.21 

27 

678.21 

27 

606.48 

28 

1048.48 

28 

886.49 

28 

767.49 

28 

676.88 

28 

605.41 

29 

1045.31 

29 

884.21 

29 

765.78 

29 

675.54 

29 

604.35 

30 

1042.14 

30 

881.95 

30 

764.08 

30 

674.22 

30 

603.29 

31 

1039.00 

31 

879.69 

31 

762.39 

31 

672.90 

31 

602.23 

32 

1035.87 

32 

877.45 

32 

760.70 

32 

671.59 

32 

601.18 

33 

1032.76 

33 

875.22 

33 

759  02 

33 

670.28 

33 

600.13 

34 

1029.67 

34 

873.00 

34 

757.35 

34 

668.98 

34 

599.09 

35 

1026.60 

35 

870.80 

35 

755.69 

35 

667.68 

35 

598.04 

36 

1023.55 

36 

868.60 

36 

754.03 

36 

666.39 

36 

597.01 

37 

1020.51 

37 

866.41 

37 

752.38 

37 

665.10 

37 

595.97 

38 

1017.49 

38 

864.24 

38 

750.74 

38 

663.82 

38 

594.94 

39 

1014.50 

39 

862.08 

39 

749.10 

39 

662.54 

39 

593.91 

40 

1011.51, 

40 

859.92 

40 

747.48 

40 

661.26 

40 

592.89 

41  1  1008.55 

41 

857.78 

41 

745.86 

41 

659.99 

41 

591.87 

4-2  \  1005.60 

42 

855.65 

42 

744.24 

42 

658.73 

42 

590.85 

43 

1002.67 

43 

853.53 

4-j 

742.63 

43 

657.47 

43 

589.84 

44 

999.76 

44 

851.42 

44 

741.03 

44 

656.22 

44 

588.83 

45 

996.87 

45 

849.32 

45 

739.44 

45 

654.97 

45 

587.83 

46 

993.99 

46 

847.23 

46 

737.86 

46 

653.72 

46 

586.82 

47 

991.13 

47 

845.15 

47 

736.28 

47 

652.48 

47 

585.83 

48 

988.28 

48 

843.08 

48 

734.70 

48 

651.25 

48 

584.83 

49 

985.45 

49 

841.02 

49 

733.14 

49 

650.02 

49 

583.84 

50 

982.64 

50 

838.97 

50 

731.58 

50 

648.79 

50 

582.85 

51 

979.84 

51 

836.93 

51 

730.03 

51 

647.57 

51 

581.86 

52 

977.06 

52 

834.90 

52 

728.48 

52 

646.35 

52 

580.88 

53 

974.29 

53 

832.89 

53 

726.94 

53 

645.14 

53 

579.90 

54 

971.54 

54 

830.88 

54 

725.41 

54 

643.94 

54 

578.92 

55 

968.81 

55 

828.88 

55 

723.88 

55 

642.73 

55 

577.95 

56 

-966.09 

56 

826.89 

56 

722.36 

56 

641.53 

56 

576  98 

57 

963.39 

57 

824.91 

57 

720.85 

57 

640.34 

57 

576.02 

58 

9G0.70 

58 

822.93 

58 

719.34 

58 

639.15 

58 

575.06 

50 

958.03 

59 

820.97 

59 

717.84 

59 

637.96 

59 

574.10 

60 

955.37 

60 

819.02 

60 

716.34 

GO 

636.78 

60 

573.14 

2U     A   FIELD-MANUAL    FOB   RAILROAD   ENGINEERS. 


TABLE  I.— RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

10°  0' 

573.14 

12°  0' 

477.68 

14°  0' 

409.32 

16°  0' 

358.17 

18°  0' 

318.39 

2 

571.24 

2 

476.36 

2 

408.35 

2 

357.43 

2 

317.80 

4 

569.35 

4 

475.05 

4 

407.38 

4 

356.69 

4 

317.22 

6 

567.47 

6 

473.74 

6 

406.42 

6 

355.95 

6 

316.63 

8 

565.60 

8 

472.44 

8 

405.46 

8 

355.21 

8 

316.05 

10 

563.75 

10 

471.15 

10 

404.51, 

10 

354.48 

10 

315.47 

12 

561.91 

12 

469.86 

12 

403.56 

12 

353.75 

12 

314.89 

14 

560.08 

14 

468.58 

14 

402.61 

14 

353.03 

14 

314.32 

16 

558.26 

16 

467.31 

16 

401.67 

16 

352.30 

16 

313.75 

18 

556.45 

18 

466.04 

18 

400.74 

18 

351.58 

18 

313.18 

20 

554.66 

20 

464.78 

20 

399.80 

20 

350.86 

20 

312.61 

22 

552.88 

22 

463.53 

22 

398.88 

2* 

350.15 

22 

312.04 

24 

551.11 

24 

462.29 

24 

397.95 

24 

349.44 

24 

311.47 

26 

549.35 

26 

461.05 

26 

397.03 

26 

348.72 

26 

310.91 

28 

547.60 

28 

459.82 

28 

396.13 

28 

348.02 

28 

310.35 

30 

545.87 

30 

458.59 

30 

395.21 

30 

347.32 

30 

309.79 

32 

544.14 

32 

457.38 

32 

394.30 

32 

346.62 

32 

309.23 

34 

542.42 

34 

456.16 

34 

393.40 

34 

345.93 

34 

308.68 

36 

540.72 

36 

454.96 

36 

392.50 

36 

345.23 

36 

808.13 

38 

539.03 

38 

453.76 

38 

391.61 

38 

344.54 

38 

307.58 

40 

537.34 

40 

452.57 

40 

390.72 

40 

343.85 

40 

307.03 

42 

5S5.67 

42 

451.38 

42 

389.83 

42 

343.16 

42 

306.48 

44 

534.01 

44 

450.20 

44 

388.95 

44 

342.48 

44 

305.93 

46 

532.36 

46 

449.02 

46 

388.07 

46 

341.80 

46 

305.39 

48 

530.71 

48 

447.86 

48 

387.20 

48 

341.12 

48 

304.85 

50 

529.08 

50 

446.69 

50 

386.33 

50 

340.45 

50 

304.31 

52 

527.46 

52 

445.54 

52 

385.47 

52 

339.78 

52 

303.77 

54 

525.85 

54 

444.39 

54 

384.60 

54 

339.11 

54 

303.24 

56 

524.25 

56 

443.24 

56 

383.75 

56 

338.44 

56 

302.70 

58 

522.65 

58 

442.11 

58 

382.89 

58 

337.77 

58 

302.17 

11°  0' 

521.07 

13°  0' 

440.97 

16°  0' 

382.04 

17°0' 

337.11 

19°  0' 

301.64 

2 

519.50 

2 

439.85 

2 

381.19 

2 

336.45 

2 

301.12 

4 

517.93 

4 

438.73 

4 

380.35 

4 

335.80 

4 

300.59 

6 

516.38 

6 

437.61 

6 

379.51 

6 

335.14 

6 

300.07 

8 

514.84 

8 

436.50 

8 

378.68 

8 

334.49 

8 

299.54 

10 

513.30 

10 

435.40 

10 

377.84 

10 

333.84 

10 

299.02 

12 

511.77 

12 

434.30 

12 

377.02 

12 

333.19 

12 

298.50 

14 

510.26 

14 

433.21 

14 

376.19 

14 

332.55 

14 

297.99 

16 

508.75 

16 

432.12 

16 

375.37 

16 

331.91 

16 

297.47 

18 

507.25 

18 

431.04 

18 

374.55 

18 

331.27 

18 

296.96 

20 

505.76 

20 

429.96 

20 

373.74 

20 

330.63 

20 

296.45 

22 

504.28 

22 

428.98 

22 

372.93 

22 

330.00 

22 

295.94 

24 

502.80 

24 

427.82 

24 

372.12 

24 

329.37 

24 

295.43 

26 

501.34 

26 

426.76 

26 

371.32 

26 

328.74 

26 

294.92 

28 

499.88 

28 

425.71 

28 

370.52 

28 

328.11 

28 

294.42 

30 

498.43 

30 

424.66 

30 

369.72 

30 

327.48 

30 

293.91 

3'> 

496.99 

32 

423.61 

32 

368.93 

32 

326.86 

32 

293.41 

34 

495.56 

34 

422.57 

34 

368.14 

34 

326.24 

34 

292.91 

36 

494.14 

36 

421.54 

36 

367.35 

30 

325.62 

36 

292.41 

38 

492.73 

38 

420.51 

38 

366.57 

38 

325.01 

38 

291.92 

40 

491.32 

40 

419.49 

40 

365.79 

40 

324.40 

40 

291.42 

42 

489.92 

42 

418.47 

42 

365.01 

42 

323.79 

42 

290.93 

44 

488.53 

44 

417.45 

44 

364.24 

44 

323.18 

44 

290.44 

46 

487.15 

46 

416.44 

46 

863.47 

46 

322.57 

46 

289.95 

48 

485.77 

48 

415.44 

48 

362.70 

48 

321.97 

48 

289.46 

50 

484.40 

50 

414.44 

50 

361.94 

50 

321.37 

50 

288.98 

5',' 

483.05 

52 

413.44 

52 

361.18 

52 

320.77 

52 

288.49 

54 

481.69 

54 

412.45 

54 

360.42 

54 

320.17 

54 

288.01 

M 

480.35 

56 

411.47 

56 

359.67 

56 

319.57 

56 

287.53 

58 

479.01 

08 

410.49 

58 

358.92 

58 

318.98 

58 

287.05 

60 

477.68 

60 

409.51 

60 

358.17 

60 

318.39 

60 

286.57 

TABLE  II.— MINUTES  IN  DECIMALS  OF  A  DEGREE.   215 


9 

0" 

10" 

15' 

20" 

30s 

40° 

45" 

50" 

/ 

0 

.00000 

.00278 

.00417 

.00556 

.00833 

.01111 

.01250 

.01389 

0 

1 

.01667 

.01944 

.02083 

.02222 

.02500 

.02778 

.02917 

.03055 

1 

2 

.03333 

.03611 

.03750 

.03889 

.04167 

.04444 

.04583 

.04722 

2 

3 

.05000 

.05278 

.05417 

.05556 

.05833 

.06111 

.06250 

.06389 

3 

4 

.06667 

.06944 

.07083 

.07222 

.07500 

.07778 

.07917 

.08056 

4 

5 

.08333 

.08611 

.08750 

.08889 

.09167 

.09444 

.09583 

.09722 

5 

6 

.10000 

.10278 

.10417 

.10556 

.10833 

.11111 

.11250 

.11389 

6 

7 

.11667 

.11944 

.12083 

.12222 

.12500 

.12778 

.12917 

.13056 

7 

8 

.13333 

.13611 

.13750 

.13889 

.14167 

.14444 

.14583 

.14722 

8 

9 

15000 

.15278 

.15417 

.15556 

.15833 

.16111 

.16250 

.16389 

9 

10 

.16667 

.16944 

.17083 

.17222 

.17500 

.17778 

.17917 

.18056 

10 

11 

.18333 

.18611 

.18750 

.18889 

.19167 

.19444 

.19583 

.19722 

11 

12 

.20000 

.20278 

.20417 

.20556 

.20833 

.21111 

.21250 

.21389 

12 

13 

.21667 

.21944 

.22083 

22222 

.22500 

.22778 

.22917 

.23056 

13 

14 

.23333 

.23611 

.23750 

!23889 

.24167 

.24444 

.24583 

.24722 

14 

15 

.25000 

.25278 

.25417 

.25556 

.25833 

.26111 

.26250 

.26389 

15 

16 

.26667 

.26944 

.27083 

.27222 

.27500 

.27778 

.27917 

.28056 

16 

17 

.28333 

.28611 

.28750 

.28889 

.29167 

.29444 

.29583 

.29722 

17 

18 

.30000 

.3027'8 

.30417 

.30556 

.30833 

.31111 

.31250 

.31389 

18 

19 

.31667 

.31944 

.32083 

.32222 

.32500 

.32778 

.32917 

.33056 

19 

20 

.33333 

.33611 

.33750 

.33889 

.34167 

.34444 

.34583 

.34722 

20 

21 

.35000 

.35278 

.35417 

.35556 

.35833 

.36111 

.36250 

.36389 

21 

22 

.36667 

.36944 

.37083 

.37222 

.37500 

.37778 

.37917 

.38056 

22 

23 

.38333 

.38611 

.38750 

.38889 

.39167 

.39444 

.39583 

.39722 

23 

24 

.40000 

.40278 

.40417 

.40556 

.40833 

.41111 

.41250 

.41389 

24 

25 

.41667 

.41944 

.42083 

.42222 

.42500 

.42778 

.42917 

.43056 

25 

26 

.43333 

.43611 

.43750 

.43889 

.44167 

J1414 

.44583 

.44722 

26 

27 

.45000 

.45278 

.45417 

.45556 

.45833 

!46111 

.46250 

.46389 

27 

28 

.46667 

.46944 

.47083 

.47222 

.47500 

.47778 

.47917 

.48056 

28 

29 

.48333 

.48611 

.48750 

.48889 

.49167 

.49444 

.49583 

.49722 

29 

30 

.50000 

.50278 

.50417 

.5U556 

.50833 

.51111 

.51250 

.51389 

30 

31 

.51667 

.51944 

.52083 

.52222 

.52500 

.52778 

.52917 

.53056 

31 

32 

.53333 

.53611 

.53750 

.53889 

.54167 

.54444 

.54583 

.54722 

32 

33 

.55000 

.55278 

.55417 

.55556 

.55833 

.56111 

.56250 

.56389 

33 

34 

.56667 

.56944 

.57083 

.57222 

.57500 

.57778 

.57917 

.58056 

34 

35 

.58333 

.58611 

.58750 

.58889 

.59167 

.59444 

.59583 

.59722 

35 

36 

.60000 

.60278 

.60417 

.60556 

.60833 

.61111 

.61250 

.61389 

36 

37 

.61667 

.61944 

.62083 

.62222 

.62500 

.62778 

.62917 

.63056 

37 

38 

.63333 

.66611 

.63750 

.63889 

.64167 

.64444 

.64583 

.64722 

38 

39 

.65000 

.65278 

.65417 

.65556 

.65833 

.66111 

.66250 

.66389 

39 

40 

.66667 

.66944 

.67083 

.67222 

.67500 

.67778 

.67917 

.68056 

40 

41 

.68333 

.68611 

.68750 

.68889 

.69167 

.69444 

.69583 

.69722 

41 

42 

.70000 

.70278 

.70417 

.70556 

.70833 

.71111 

.71250 

.71389 

42 

43 

.71667 

.71944 

.72083 

722^2 

.72500 

.72778 

.72917 

.73056 

43 

44 

.73333 

.73611 

.73750 

.'73889 

.74167 

.74444 

.74583 

.74722 

44 

45 

.75000 

.75278 

.75417 

.75556 

.75833 

.76111 

.76250 

.76389 

45 

46 

.76667 

.76944   .77083 

.77222 

.77500   .77778 

.77917 

.78056 

46 

47 

.78333 

.78611 

.78750 

.78889 

.79167 

.79444 

.79583 

.79722 

47 

48 

.80000 

.80278 

.80417 

.80556 

.80833 

.81111 

.81250 

.81389 

48 

49 

.81667 

.81944 

.82083 

.82222 

.82500 

.82778 

.82917 

.83056 

49 

50 

.83333 

.83611 

.83750 

.&S889 

.84167 

.84444 

.84583 

.84722 

50 

51 

.85000 

.85278 

.85417 

.85556 

.85833 

86111 

.86250 

.86389 

51 

52 

.86667 

.86944 

.87083 

.87222 

.87500 

.87778 

.87917 

.88056 

52 

53 

.88333 

.88611 

.88750 

.88889 

.89167 

.89444 

.89583 

.89722 

53 

54 

.90000 

.90278 

.90417 

.90556 

.90833 

.91111 

.91250 

.91389 

54 

55 

.91667 

.91944 

.92083 

.92222 

.92500 

.92778 

.92917 

.93056 

55 

56 

.93333 

.93611 

.93750 

.93889 

.94167 

94444 

.94583 

.94722 

56 

57 

.95000 

.95278 

.95417 

.95556 

.95883 

!96111 

.96250 

.96389 

57 

58 

.96667 

.96944 

.97083 

97222 

.97500 

.97778 

.97917 

.98056 

58 

59 

.98333 

.98611 

.98750 

98889 

.99167 

.99444 

99583 

.99722 

59 

/ 

0" 

10' 

15" 

20" 

30" 

40" 

45" 

50" 

/ 

216     A   FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 


o 


09   I  00  00  00  00  30  00  00  C 

1-1   <N  ii  oi  ci  si  •?>  <?*  c 


<ri  <N ci lot  oi  st  c\j  cj  oi  T« 


.-  oo  oo  oo  oo  oc  oo 


O*  CJ  O»  OJ  C*  S*  O»  <N  C} 


i— <     C?  C*  CO  ^J*  lO  iO  CO  t—  t'-  OO 
C)9*etC*0<CH     (NfftCJiNlNOi^CJlNO* 


»n  »o  «o  »o  10  »n  »o  o  10  in    «n  m  m  10  »o  m  co  o  «s  cc 


iiO  IQM  «O  O 


O»»T3QO^-Oi3«OO»<N»rtOO»-i 

o  co  «o  i-  i-  1-  1-  ao  ao  ao  o 


ooeoeocococococococo 


to*    <NC*C*OJNC»WCJC*C»   emstvietcmcicict 


00000000000  0000000000  OOOOOOOOOO 


TABLES. 


217 


TABLE   IV.— LONG   CHORDS. 


Degree  of 
Curve. 

Actual  Arc, 
One  Station. 

Long  Chords. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 
Stations. 

0° 

10' 

100.000 

200.00 

300.00 

400.00 

500.00 

599.99 

20 

.000 

200.00 

300.00 

399.99 

499.98 

599.97 

30 

.000 

200.00 

299.99 

399.98 

499.96 

599.93 

40 

.001 

200.00 

299.99 

399.97 

499.93 

599.88 

50 

.001 

200.00 

299.98 

399.95 

499.89 

599.82 

1 

100.001 

199.99 

299.97 

399.92 

499.85 

599.73 

10 

.002 

199.99 

299.96 

399.90 

499.79 

599.64 

20 

002 

199.99 

299.95 

399.87 

499.73 

599.53 

30 

.003 

199.98 

299.93 

399.83 

499.66 

599.40 

40 

.003 

199.98 

299.92 

399.79 

499.58 

599.26 

50 

.004 

199.97 

299.90 

399.74 

499.49 

599.11 

2 

100.005 

199.97 

299.88 

399.70 

499.39 

598.93 

10 

.006 

199.96 

299.86 

399.64 

499.29 

598.75 

20 

.007 

199.96 

299.83 

399.59 

499.17 

598.55 

80 

.003 

199.95 

299.81 

399.52 

499.05 

598.34 

40 

.009 

199.95 

299.78 

399.46 

498.92 

598.11 

50 

.010 

199.94 

299.76 

399.39 

498.78 

597.86 

3 

100.011 

199.93 

299.73 

399.32 

498.63 

597.60 

10 

.013 

199.92 

299.70 

399.24 

498.47 

597.33 

20 

.014 

199.92 

299.66 

399.15 

498.31 

597.04 

30 

.015 

199.91 

299.63 

399.07 

498.14 

596.74 

40 

.017 

199.90 

299.59 

398.98 

497.96 

596.42 

50 

.019' 

199.89 

299.55 

398.88 

497.77 

596.09 

4 

100.020 

199.88 

299.51 

398.78 

497.57 

595.74 

10 

.022 

199.87 

299.47 

398.68 

497.36 

595.38 

20 

.024 

199.86 

299.43 

398.57 

497.15 

595.01 

30 

.026 

199.85 

299.38 

398.46 

496.92 

594.62 

40 

.028 

199.83 

299.34 

398.34 

496.69 

594.21 

50 

.030 

199.82 

299.29 

398.22 

496.45 

593.79 

5 

100.032 

199.81 

299.24 

398.10 

496.20 

593.36 

10 

.0-34 

199.80 

299.19 

397.97 

495.94 

592.91 

20 

.036 

199.78 

299.13 

397.84 

495.68 

592.45 

30 

.038 

199.77 

299.08 

397.70 

495.41 

591  .97 

40 

.041 

199.76 

299.02 

397.56 

495.12 

591.48 

50 

.043 

199.74 

298.96 

397.41 

494.83 

590.97 

6 

100.046 

199.73 

298.90 

397.26 

494.53 

590.45 

10 

.048 

199.71 

298.84 

397.11 

494.23 

589.91 

20 

.051 

199.70 

298.78 

396.95 

493.91 

589.36 

30 

.054 

199.68 

298.71 

396.79 

493.59 

588.80 

40 

.056 

199.66 

298.65 

396.62 

493.26 

588.22 

50 

.059 

199.64 

298.58 

396.45 

492.92 

587.63 

1 

100.062 

199.63 

298.51 

396.28 

492.57 

587.02 

10 

.016 

199.51 

298.30 

395.91 

491.97 

586.12 

20 

.017 

199.49 

298.21 

395.71 

491.60 

585.46 

30 

.018 

199.47 

298.13 

395.52 

491.21 

584.80 

40 

.018 

199.44 

298.05 

395.32 

490.82 

584.13 

50 

.019 

199.42 

297.96 

395.11 

490.42 

583.44 

8 

100.020 

199.39 

297.87 

394.90 

490.01 

582.72 

10 

.021 

199.37 

297.78 

394.69 

489.59 

582.01 

20 

.022 

199.34 

297.69 

394.47 

489.16 

581.27 

30 

.023 

199.31 

297.60 

394.25 

488.73 

580.52 

40 

.024 

199.28 

297.50 

394.02 

488.28 

579.76 

50 

.025 

199.26 

297.41 

393.79 

487.83 

578.98 

9 

100.026 

199.23 

297.31 

393.55 

487.37 

578.18 

10 

.027 

199.20 

297.21 

393.31 

486.90 

577.38 

20 

.028 

199.17 

297.10 

393.07 

486.43 

576.56 

30 

.029 

199.14 

297.00 

392.82 

485.95 

575.73 

40 

.030 

199.11 

296.90 

392.57 

485.45 

574.88 

50 

.031 

199.08 

296.79 

392.31 

484.95 

574.02 

10 

100.032 

199.05 

296.68 

392.05 

484.44 

573.14 

218     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS, 


TABLE  IV.— LONG  CHORDS. 


Degree 
of 
Curve. 

Actual  Arc, 
One 
Station. 

Long  Chords. 

1 
Station. 

2 
Stations. 

3                  4 
Stations.  Stations. 

5 
Stations. 

10°     0' 

100.032 

99.91 

199  05 

296.68 

392.05 

484.44 

10 

.033 

99.90 

199.02 

296.57 

391.79 

483.92 

20 

.034 

99.90 

198.98 

296.45 

391.51 

483.39 

30 

.035 

99.90 

198.95 

296.33 

391.24 

482.86 

40 

.036 

99.89 

198.92 

296.22 

390.96 

482.32 

50 

.037 

99.89 

198.88 

296.10 

390.68 

481.77 

11 

100.038 

99.88 

198.85 

295.98 

390.39 

481.21 

10 

.040 

99.88 

198.81 

295.86 

390.10 

480.64 

20 

.041 

99.88 

198.78 

295.74 

389.81 

480.07 

30 

.042 

99.87 

198.74 

295.61 

389.50 

479.48 

40 

.043 

99.87 

198.71 

295.48 

389.20 

478.89 

50 

.044 

99.87 

198.67 

295.35 

388.89 

478.29 

12 

100.046 

99.86 

198.63 

295.22 

388.58 

477.68 

10 

.047 

99.86 

198.59 

295.09 

388.27 

477.07 

20 

.048 

99.85 

198.55 

294.95 

387.95 

476.44 

30 

.050 

99.85 

198.51 

294.82 

387.62 

475.81 

40 

.051 

99.85 

198.48 

294.68 

387.29 

475.18 

50 

.052 

99.84 

198.43 

294.54 

386.95 

474.52 

13 

100.054 

99.84 

198.39 

294.40 

386.62 

473.87 

10 

.055 

99.84 

198.35 

294.26 

386.28 

473.20 

20 

.056 

99.83 

198.31 

294.11  ' 

385.93 

472.53 

30 

.058 

99.83 

198.27 

293.96 

385.58 

471.86 

40 

.059 

99.82 

198.23 

293.81 

385.23 

471.17 

50 

.061 

99.82 

198.18 

293.66 

384.87 

470.48 

14 

100.062 

99.81 

198.14 

293.51 

384.51 

469.77 

TABLE   V.— MID-ORDINATES   TO   LONG   CHORDS. 


Degree  of 
Curve. 

1 
Station. 

2 

Stations. 

3 
Stations. 

4 
Stations. 

5 
Stations. 

6 
Stations. 

0°  10' 

.04 

.15 

.33 

.58 

.91 

1.31 

20 

.07 

.29 

.65 

1.16 

1.82 

2.62 

30 

.11 

.44 

.98 

1.75 

2.73 

3.93 

40 

.15 

.58 

1.31 

2.33 

3.64 

5.24 

50 

.18 

.73 

1.64 

2.91 

4.55 

6.54 

1 

.22 

.87 

1.96 

3.49 

5.45 

7.85 

10 

.26 

.02 

2.29 

4.07 

6.36 

9.16 

20 

.29 

.16 

2.62 

4.65 

7.27 

10.47 

30 

.33 

.31 

2.95 

5.24 

8.18 

11.78 

40 

.36 

.45 

3.27 

5.82 

9.09 

13.08 

50 

.40 

.60 

3.60 

6.40 

9.99 

14.39 

2 

.44 

-   *  •) 

3.93 

6.98 

10.90 

15  69 

10 

.47 

.89 

4.25 

7.56 

11.81 

17.00 

20 

.51 

2.04 

4.58 

8.14 

12.72 

18.30 

30 

.55 

2.18 

4.91 

8.72 

13.62 

19.61 

40 

.58 

2.33 

5.23 

9.30 

14.53 

20.91 

50 

.62 

2.47 

5.56 

9.88 

15.44 

22.21 

3 

.65 

2.62 

5.89 

10.46 

16.34 

23.52 

10 

.69 

2.76 

6.22 

11.04 

17.25 

24.82 

20 

.73 

'      2.91 

6.54 

11.62 

18.15 

26.12 

30 

.76 

3.05 

6.87 

12.20 

19.06 

27.42 

40 

.80 

3.20 

7.20 

12.78 

19.96 

28.71 

50 

.84 

3.35 

7.52 

13.36 

20.86 

30.01 

4 

.87 

3.49 

7.85 

13.94 

21.77 

31.31 

TABLES. 


219 


TABLE  V.— MID-ORDINATES   TO   LONG   CHORDS. 


Degree  of 
Curve. 

1 
Station. 

2 
Stations. 

3 
Stations. 

4 
Stations. 

5 
Stations. 

6 

Stations. 

4°     0' 

.87 

3.49 

7.85 

13.94 

21  .  77 

31.31 

10 

.91 

3.64 

8.18 

14.52 

oo  (57 

32.60 

20 

.95 

3.78 

8.50 

15.10 

23^57 

33.90 

30 

.98 

3.93 

8.83 

15.68 

24.47 

35.19 

40 

1.02 

4.07 

9.15 

16.26 

25.37 

36.48 

50 

1.05 

4  22 

9.48 

16.84 

26.27 

37.77 

5 

1.09 

4.36 

9.81 

17  42 

27.17 

39.06 

10 

1.18 

4.51 

10.13 

17.99 

28.07 

40.35 

20 

1.16 

4.65 

10.46 

18.57 

28.97 

41.63 

30 

1.20 

4.80 

10.79 

19.15 

29.87 

42.92 

40 

1.24 

4.94 

11.11 

19.72 

30.76 

44.20 

50 

1.27 

5.09 

11.44 

20.30 

31.66 

45.48 

6 

1.31 

5.23 

11.76 

20.88 

32.55 

46.76 

10 

1.35 

5.38 

12.09 

21.45 

33.45 

48.04 

20 

1.38 

5.52 

12.41 

22  03 

34.34 

49.31 

30 

1.42 

5.67 

12.74 

22.60 

35.23 

50.59 

40 

1.46 

5.81 

13.06 

23.18 

36.13 

51.86 

50 

1.49 

5.96 

13.39 

23.75 

37.02 

53.13 

7 

1.53 

6.11 

13.72 

24.33 

37.91 

54.40 

10 

1.56 

6.25 

14.03 

24.89 

38.79 

55.64 

20 

1.60 

6.39 

14.36 

25.46 

39.66 

56.90 

30 

1.64 

6.54 

14.68 

26.04 

40.55 

58.16 

40 

1.67 

6.68 

15.01 

26.61 

41.43 

59.41 

r>0 

1.71 

6.83 

15.33 

27.18 

42.32 

60.68 

8 

1.74 

6.97 

15.65 

**7  75 

43.20 

61.93 

10 

1.78 

7.12 

15.98 

28^32 

44.08 

63.18 

20 

1.82 

7.26 

16.30 

28.89 

44.96 

64.43 

30 

1.85 

7.41 

16.62 

29.46 

45.84 

65.68 

40 

1.89 

7.55 

16.95 

30.03 

46.72 

66.92 

50 

1.93 

7.70 

17.27 

30.60 

47.60 

68'.  17 

9 

1.96 

7.84 

17.59 

31.17 

48.47 

69.40 

10 

2.00 

7.98 

17.92 

31.73 

49.35 

70.64 

20 

2.04 

8.13 

18.24 

32.30 

50.22 

71.88 

30 

2.07 

8.27 

18.56 

32.87 

51.09 

73.11 

40 

2.11 

8.42 

18  88 

33.43 

51.96 

74.34 

50 

2.14 

8.56 

19.21 

34.00 

52.83 

75.56 

10 

2.18 

8.71 

19.53 

34.56 

53.70 

76.79 

10 

2  22 

8.85 

19.85 

35.13 

54.57 

20 

2.25 

9.00 

20.17 

35.69 

55.43 

30 

2.29 

9.14 

20.49 

36-26 

56.29 

40 

2.33 

9.28 

20.82 

36.82 

57.16 

50 

2.36 

9.43 

21.14 

37.38 

58.02 

11 

2.40 

9.57 

21.46 

37.94 

58.88 

10 

2.44 

9.72 

21.78 

38.50 

59.73 

20 

2.47 

9.86 

22.10 

39  06 

60.58 

30 

2.51 

10.01 

22.42 

39.62 

61.44 

40 

2.54 

10.15 

22.74 

40.18 

62.30 

50 

2.58 

10.29 

23.06 

40.74 

63.15 

12 

2  62 

10.44 

23.38 

41.30 

64.00 

10 

2.65 

10.58 

23.70 

41.86 

64.85 

20 

2.69 

10.73 

24.02 

42.41 

65.69 

- 

30 

2.73 

10.87 

24.34 

42.97 

66.54 

40 

2.76 

11.01 

24.66 

43.52 

67.38 

50 

2.80 

11.16 

24.97 

44.08 

68.22 

13 

2.83 

11.30 

25.29 

44.63 

69.06 

10 

2.87 

11.45 

25.61 

45.18 

69.90 

20 

2.91 

11.59 

25.93 

45.73 

70.73 

30 

2.94 

11.73 

26.25 

46.29 

71.57 

40 

2.98 

11.88 

26.57 

46.84 

72.40 

50 

3.02 

12.02 

26.88 

47.39 

73.23 

14                   3.05 

12.16 

27.20 

47.93 

74.06 

220  TABLE  VI.— LOGARITHMS   OF  NUMBERS. 


100 

1 
2 
3 

4 
5 
6 

7 


110 

1 
2 
3 
4 
5 
6 
7 
8 
9 

120 

1 
2 
3 
4 
5 
6 
7 


130 

1 
2 
3 

4 
5 
6 
7 
8 
9 

140 

1 
2 
3 
4 
5 
6 
7 
8 
9 

150 


6789 


00000  00043  00087  00130  00173  00217  00260  00303  00340  00389 

0432  0475  0518  0561  0604  0647  0689  0732  0775  0817 

0860  0903  0945  0988  1030  1072  1115  1157  1199  1242 

1284  1326  1368  1410  1452  1494  1536  1578  1620  1662 

1703  1745  1787  1828  1870  1912  1953  1995  2036  2078 

2119  2160  2202  2243  2284  2325  2366  2407  2449  2490 

2531  2572  2612  2653  2694  2735  2776  2816  2857  2898 

2938  2979  3019  3060  3100  3141  3181  3222  3262  3302 

3342  3383  3423  3463  3503  3543  3583  3623  3663  3703 

3743  3782  3822  3882  3902  3941  3981  4021  4060  4100 

04139  04179  04218  04258  04297  04336  04376  04415  04454  04493 

4532  4571  4610  4650  4689  4727  4766  4805  4844  4883 

4922  4961  4999  5038  5077  5115  5154  5192  5231  5269 

5308  5346  5385  5423  5461  5500  5538  5576  5614  5652 

5690  5729  5767  5805  5843  5881  5918  5956  5994  6032 

6070  6108  6145  6183  6221  6258  6296  6333  6371  6408 

6446  6483  6521  6558  6595  6633  6670  6707  6744  6781 

6819  6856  6893  6930  6967  7004  7041  7078  7115  7151 

7188  7225  7262  7298  7335  7372  7408  7445  7482  7518 

7555  7591  7628  7664  7700  7737  7773  7809  7846  7882 

07918  07954  07990  08027  08063  08099  08135  08171  08207  08243 

8279  8314  8350  8386  8422  8458  8493  8529  8565  8600 

8636  8672  8707  8743  8778  8814  8849  8884  8920  8955 

8991  9026  9061  9096  9132  9167  9202  9237  9272  9307 

9342  9377  9412  9447  9482  9517  9552  9587  9621  9656 

9691  9726  9760  9795  9830  9864  9899  9934  9968  10003 

10037  10072  10106  10140  10175  10209  10243  10278  10312  0346 

0380  0415  0449  0483  0517  0551  0585  0619  0653  0687 

0721  0755  0789  0823  0857  0890  0924  0958  0992  1025 

1059  1093  1126  1160  1193  1227  1261  1294  1327  1361 

11394  11428  11461  11494  11528  11561  11594  11628  11661  11694 

1727  1760  1793  1826  1860  1893  1926  1959  1992  2024 

205?  2090  2123  2156  2189  2222  2254  2287  2320  2352 

2385  2418  2450  2483  2516  2548  2581  2613  2646  2678 

2710  2743  2775  2808  2840  2872  2905  2937  2969  3001 

3033  3066  3098  3130  3162  3194  3226  3258  3290  3322 

3354  3386  3418  3450  3481  3513  3545  3577  3609  3640 

3672  3704  3735  3767  3799  3830  3862  3893  3925  3956 

3988  4019  4051  4082  4114  4145  4176  4208  4239  4270 

4301  4333  4364  4395  4426  4457  4489  4520  4551  4582 

14613  14644  14675  14706  14737  14768  14799  14829  14860  14891 

4922  4953  4983  5014  5045  5076  5106  5137  5168  5198 

5229  5259  5290  5320  5351  5381  5412  5442  5473  5503 

5534  5564  5594  5625  5655  5685  5715  5746  5776  5806 

5836  5866  5897  5927  5957  5987  6017  6047  6077  6107 

6137  6167  6197  6227  6256  6286  6316  6346  6376  6406 

6435  6465  6495  6524  6554  6584  6613  6643  6673  6702 

6732  6761  6791  6820  6850  6879  6909  6938  6967  (J997 

7026  7056  7085  7114  7143  7173  7202  7231  7200  7289 

7319  7348  7377  7406  7435  7464  7493  7522  7551  7580 

17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 


TABLE  VI.— LOGARITHMS  OP  NUMBERS. 


0123456789 


150  17609  17638  17667  17696  17725.  17754  17782  17811  17840  17869 

1  7898  7926  7955  7984  8013  8041  8070  8099  8127  8156 

2  8184  8213  8241  8270  8298  8327  8355  8384  8412  8441 

3  8469  8498  8526  8554  8583  8611  8639  8667  8696  8724 

4  8752  8780  8808  8837  8865  8893  8921  8949  8977  9005 

5  9033  9061  9089  9117  9145  9173  9201  9229  9257  9285 

6  9312  9340  9368  9396  9424  9451  9479  9507  9535  9562 

7  9590  9618  9645  9673  9700  9728  9756  9783  9811  9838 

8  9866  9893  9921  9948  9976  20003  20030  20058  20085  20112 

9  20140  20167  20194  20222  20249  0276  0303  0330  0358  0385 

160  20412  20439  20466  20493  20520  20548  20575  20602  20629  20656 

1  0683  0710  0737  0763  0790  0817  0844  0871  0898  0925 

2  0952  0978  1005  1032  1059  1085  1112  1139  1165  1192 

3  1219  1245  1272  1299  1325  1352  1378  1405  1431  1458 

4  1484  1511  1537  1564  1590  1617  1643  1669  1696  1¥22 

5  1748  1775  1801  1827  1854  1880  1906  1932  1958  1985 

6  2011  2037  2063  2089  2115  2141  2167  2194  2220  2246 

7  2272  2298  2324  2350  2376  2401  2427  2453  2479  2505 

8  2531  2557  2583  2608  2634  2660  2686  2712  2737  2763 

9  2789  2814  2840  2866  2891  2917  2943  2968  2994  3019 

170  23045  23070  23096  23121  23147  23172  23198  23223  23249  23274 

1  3300  3325  3350  3376  3401  3426  3452  3477  3502  3528 

2  3553  3578  3603  3629  3654  3679  3704  3729  3754  3779 

3  3805  3830  3855  3880  3905  3930  3955  3980  4005  4030 

4  4055  4080  4105  4130  4155  4180  4204  4229  4254  4279 

5  4304  4329  4353  4378  4403  4428  4452  4477  4502  4527 

6  4551  4576  4601  4625  4650  4674  4699  4724  4748  4773 

7  4797  4822  4846  4871  4895  4920  4944  4969  4993  5018 

8  5042  5066  5091  5115  5139  5164  5188  5212  5237  5261 

9  5285  5310  5334  5358  5382  5406  5431  5455  5479  5503 

180  25527  25551  25575  25600  25624  25648  25672  25696  25720  25744 

1  5768  5792  5816  5840  5864  5888  5912  5935  5959  5983 

2  6007  6031  6055  6079  6102  6126  6150  6174  6198  6221 

3  6245  6269  6293  6316  6340  6364  6387  6411  6435  6458 

4  6482  6505  6529  6553  6576  6600  6623  6647  6670  6694 

5  6717  6741  6764  6788  6811  6834  6858  6881  6905  6928 

6  6951  6975  6998  7021  7045  7068  7091  7114  7138  7161 

7  7184  7207  7231  7254  7277  7300  7323  7346  7370  7393 

8  7416  7439  7462  7485  7508  7531  7554  7577  7600  7623 

9  7646  7669  7692  7715  7738  7761  7784  7807  7830  7852 

190  27875  27898  27921  27944  27967  27989  28012  28035  28058  28081 

1  8103  8126  8149  8171  8194  8217  8240  8262  8285  8307 

2  8330  8353  8375  8398  8421  8443  8466  8488  8511  8533 

3  8556  8578  8601  8623  8646  8668  8691  8713  8735  8758 

4  8780  8803  8825  8847  8870  8892  8914  8937  8959  8981 

5  9003  9026  9048  9070  9092  9115  9137  9159  9181  9203 

6  9226  9248  9270  9292  9314  9336  9358  9380  9403  9425 

7  9447  9469  9491  9513  9535  9557  9579  9601  9623  9645 

8  9667.  9688  9710  9732  9754  9776  9798  9820  9842  98(53 

9  9885  9907  9929  9951  9973  9994  30016  30038  30060  30081 

200  30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 


222 


TABLE  VI.— LOGARITHMS  OF  NUMBERS. 


'H 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200 

30103 

30125 

30146 

30168 

30190 

30211 

30233 

30255 

30276 

30298 

1 

0320 

0341 

0363 

0384 

0406 

0428 

0449 

0471 

0492 

0514 

2 

0535 

0557 

0578 

0600 

0621 

0643 

0664 

0685 

0707 

0728 

3 

0750 

0771 

0792 

0814 

0835 

0856 

0878 

0899 

0920 

0942 

4 

0963 

0984 

1006 

1027 

1048 

1069 

1091 

1112 

1133 

1154 

5 

1175 

1197 

1218 

1239 

1260 

1281 

1302 

1323 

1345 

1366 

6 

1387 

1408 

1429 

1450 

1471 

1492 

1513 

1534 

1555 

1576 

7 

1597 

1618 

1639 

1660 

1681 

1702 

1723 

1744 

1765 

1785 

8 

1806 

1827 

1848 

1869 

1890 

1911 

1931 

1952 

1973 

1994 

9 

2015 

2035 

2056 

2077 

2098 

2118 

2139 

2160 

2181 

2201 

210 

32222 

32243 

32263  32284 

32305 

32325 

32346 

32366 

32387 

32408 

1 

2428 

2449 

2469 

2490 

2510 

2531 

2552 

2572 

2593 

2613 

2 

2634 

2654 

2675 

2695 

2715 

2736 

2756 

2777 

2797 

2818 

3 

2838 

2858 

2879 

2899 

2919 

2940 

2960 

2980 

3001 

3021 

4 

3041 

3062 

3082 

3102 

3122 

3143 

3163 

3183 

3203 

3224 

5 

3244 

3264 

3284 

3304 

3325 

3345 

3365 

3385 

3405 

3425 

6 

3445 

3465 

3486 

3506 

3526 

3546 

3566 

3586 

3606 

3626 

7 

3646 

3666 

3686 

3706 

3726 

3746 

3766 

3786 

3806 

3826 

8 

3846 

3866 

3885 

3905 

3925 

3945 

3965 

3985 

4005 

4025 

9 

4044 

4064 

4084 

4104 

4124 

4143 

4163 

4183 

4203 

4223 

220 

34242  34262  34282 

34301 

34321 

34341 

34361 

34380 

34400 

34420 

1 

4439 

4459 

4479 

4498 

4518 

4537 

4557 

4577 

4596 

4616 

2 

4635 

4655 

4674 

4694 

4713 

4733 

4753 

4772 

4792 

4811 

3 

4830 

4850 

4869 

4889 

4908 

4928 

4947 

4967 

4986 

5005 

4 

5025 

5044 

5064 

5083 

5102 

5122 

5141 

5160 

5180 

5199 

5 

5218 

5238 

5257 

5276 

5295 

5315 

5334 

5353 

5372 

5392 

6 

5411 

5430 

5449 

5468 

5488 

5507 

5526 

5545 

5564 

5583 

7 

5603 

5622 

5641 

5660 

5679 

5698 

5717 

5736 

5755 

5774 

8 

5793 

5813 

5832 

5851 

5870 

5889 

5908 

5927 

5946 

5965 

9 

5984 

6003 

6021 

6040 

6059 

6078 

6097 

6116 

6135 

6154 

280 

36173 

36192  36211 

36229  36248 

36267 

36286  36305 

36324  36342 

1 

6361 

6380 

6399 

6418 

6436 

6455 

6474 

6493 

6511 

6530 

2 

6549 

6568 

6586 

6605 

6624 

6642 

6661 

6680 

6698 

6717 

3 

6736 

6754 

6773 

6791 

6810 

6829 

6847 

6866 

6884 

6903 

4 

6922 

6940 

6959 

6977 

6996 

7014 

7033 

7051 

7070 

7088 

5 

7107 

7125 

7144 

7162 

7181 

7199 

7218 

7236 

7254 

7273 

6 

7291 

7310 

7328 

7346 

7365 

7383 

7401 

7420 

7438 

7457 

7 

7475 

7493 

7511 

7530 

7548 

7566 

7585 

7603 

7621 

7639 

8 

7658 

7676 

7694 

7712 

7731 

7749 

7767 

7785 

7803 

7822 

9 

7840 

7858 

7876 

7894 

7912 

7931 

7949 

7967 

7985 

8003 

240 

38021 

38039  38057 

38075 

38093 

38112 

38130 

38148 

38166 

38184 

1 

8202 

8220 

8238 

8256 

8274 

8292 

8310 

8328 

8346 

8364 

2 

8382 

8399 

8417 

8435 

8453 

8471 

8489 

8507 

8525 

8543 

3 

8561 

8578 

8596 

8614 

8632 

8650 

8668 

8686 

8703 

8721 

4 

8739 

8757 

8775 

8792 

8810 

8828 

8846 

8863 

8881 

8899 

5 

8917 

8934 

8952 

8970 

8987 

9005 

9023 

9041 

9058 

9076 

6 

9094 

9111 

9129 

9146 

9164 

9182 

9199 

9217 

9235 

9252 

7 

9270 

9287 

9305 

9322 

9340 

9358 

9375 

9393 

9410 

9428 

8 

9445 

9463 

9480 

9498 

9515 

9533 

9550 

9568 

9585 

9602 

9 

9620 

9637 

9655 

9672 

9690 

9707 

9724 

9742 

9759 

9777 

250 

39794 

39811 

39829  39846 

39863 

39881 

39898 

39915 

39933 

39950 

TABLE  VI.— LOGARITHMS   OP   NUMBERS.  223 


6789 


250 

1 
2 
3 

4 
5 
6 
7 
8 
9 

260 

1 
2 
3 

4 
5 
6 
7 
8 
9 

270 

1 
2 
3 

4 
5 
6 
7 
8 
9 

280 

1 
2 
3 

4 
5 
6 
7 
8 
9 

290 

1 
2 
3 
4 
5 
6 
7 
8 
9 

300 


39794  39811  39829  39846  39863  39881  39898  39915  39933  39950 

9967  9985  40002  40019  40037  40054  40071  40088  40106  40123 

4014040157  0175  0192  0209  0226  0243  0261  0278  0295 

0312  0329  0346  0364  0381  0398  0415  0432  0449  0466 

0483  0500  0518  0535  0552  0569  0586  0603  0620  0637 

0654  0671  0688  0705  0722  0739  0756  0773  0790  0807 

0824  0841  0858  0875  0892  0909  0926  0943  0960  0976 

0993  1010  1027  1044  1061  1078  1095  1111  1128  1145 

1162  1179  1196  1212  1229  1246  1263  1280  1296  1313 

1330  1347  1363  1380  1397  1414  1430  1447  1464  1481 

41497  41514  41531  41547  41564  41581  41597  41614  41631  41647 

1664  1681  1697  1714  1731  1747  1764  1780  1797  1814 

1830  1847  1863  1880  1896  1913  1929  1946  1963  1979 

"1996  2012  2029  2045.2062  2078  2095  2111  2127  2144 

2160  2177  2193  2210  2226  2243  2259  2275  2292  2308 

2325  2341  2357  2374  2390  2406  2423  2439  2455  2472 

2488  2504  2521  2537  2553  2570  2586  2602  2619  2635 

2651  2667  2684  2700  2716  2732  2749  2765  2781  2797 

2813  2830  2846  2862  2878  2894  2911  2927  2943  2959 

2975  2991  3008  3024  3040  3056  3072  3088  3104  3120 

43136  43152  43169  43185  43201  43217  43233  43249  43265  43281 

3297  3313  3329  3345  3361  3377  3393  3409  3425  3441 

3457  3473  3489  3505  3521  3537  3553  3569  3584  3600 

3616  3632  3648  3664  3680  3696  3712  3727  3743  3759 

3775  3791  3807  3823  3838  3854  3870  3886  3902  3917 

3933  3949  3965  3981  3996  4012  4028  4044  4059  4075 

4091  4107  4122  4138  4154  4170  4185  4201  4217  4232 

4248  4264  4279  4295  4311  4326  4342  4358  4373  4389 

4404  4420  4436  4451  4467  4483  4498  4514  4529  4545 

4560  4576  4592  4607  4623  4638  4654  4669  4685  4700 

44716  44731  44747  44762  44778  44793  44809  44824  44840  44855 

4871  4886  4902  4917  4932  4948  4963  4979  4994  5010 

5025  5040  5056  5071  5086  5102  5117  5133  5148  5163 

5179  5194  5209  5225  5240  5255  5271  5286  5301  5317 

5332  5347  5362  5378  5393  5408  5423  5439  5454  5469 

5484  5500  5515  5530  5545  5561  5576  5591  5606  5621 

5637  5652  5667  5682  5697  5712  5728  5743  5758  5773 

5788  5803  5818  5834  5849  5864  5879  5894  5909  5924 

5939  5954  5969  5984  6000  6015  6030  6045  6060  6075 

6090  6105  6120  6135  6150  6165  6180  6195  6210  6225 

46240  46255  46270  46285  46300  46315  46330  46345  46359  46374 

6389  6404  6419  6434  6449  6464  6479  6494  6509  6523 

6538  6553  6568  6583  6598  6613  6627  66*2  6657  6672 

6687  6702  6716  6731  6746  6761  6776  6790  6805  6820 

6835  6850  6864  6879  6894  6909  6923  6938  6953  6967 

6982  6997  7012  7026  7041  7056  7070  7085  7100  7114 

7129  7144  7159  7173  7188  7202  7217  7232  7246  7261 

7276  7290  7305  7319  7334  7349  7363  7378  7392  7407 

7422  7436  7451  7465  7480  7494  7509  7524  7538  7.553 

7567  7582  7596  7611  7625  7640  7654  7669  7683  7698 

47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 


224  TABLE  VI.— LOGARITHMS  OP  NUMBERS. 


300 

1 
2 
3 
4 
5 
6 
7 
8 
9 

310 

1 

2 
3 
4 
5 
6 
7 
8 
9 

320 

1 
2 
3 
4 
5 
6 
7 
8 
9 

330 

1 

2 
3 

4 
5 
6 
7 
8 
9 

340 

1 
2 
3 
4 
5 
6 
7 
8 
9 

350 


0123456789 


47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 

7857  7871  7885  7900  7914  7929  7943  7958  7972  7986 

8001  8015  8029  8044  8058  8073  8087  8101  8116  8130 

8144  8159  8173  8187  8202  8216  8230  8244  8259  8273 

8287  8302  8316  8330  8344  8359  8373  8387  8401  8416 

8430  8444  8458  8473  8487  8501  8515  8530  8544  8558 

8572  8586  8601  8615  8629  8643  8657  8671  8686  8700 

8714  8728  8742  8756  8770  8785  8799  8813  8827  8841 

8855  8869  8883  8897  8911  8926  8940  8954  8968  8982 

8996  9010  9024  9038  9052  9066  9080  9094  9108  9122 

49136  49150  49164  49178  49192  49206  49220  49234  49248  49262 

9276  9290  9304  9318  9332  9346  9360  9374  9388  9402 

9415  9429  9443  9457  9471  9485  9499  9513  9527  9541 

9554  9568  9582  9596  9610  9624  9638  9651  9665  9679 

9693  9707  9721  9734  9748  9762  9776  9790  9803  9817 

9831  9845  9859  9872  9886  9900  9914  9927  9941  9955 

9969  9982  9996  50010  50024  50037  50051  50065  50079  50092 

50106  50120  50133  0147  0161  0174  0188  0202  0215  0229 

0243  0256  0270  0284  0297  0311  0325  0338  0352  0365 

0379  0393  0406  0420  0433  0447  0461  0474  0488  0501 

50515  50529  50542  50556  50569  50583  50596  50610  50623  50637 

0651  0664  0678  0691  0705  0718  0732  0745  0759  0772 

0786  0799  0813  0826  0840  0853  0866  0880  0893  0907 

0920  0934  0947  0961  0974  0987  1001  1014  1028  1041 

1055  1068  1081  1095  1108  1121  1135  1148  1162  1175 

1188  1202  1215  1228  1242  1255  1268  1282  1295  1308 

1322  1335  1348  1362  1375  1388  1402  1415  1428  1441 

1455  1468  1481  1495  1508  1521  1534  1548  1561  1574 

1587  1601  1614  1627  1640  1654  1667  1680  1693  1706 

1720  1733  1746  1759  1772  1786  1799  1812  1825  1838 

51851  51865  51878  51891  51904  51917  51930  51943  51957  51970 

1983  1996  2009  2022  2035  2048  2061  2075  2088  2101 

2114  2127  2140  2153  2166  2179  2192  2205  2218  2231 

2244  2257  2270  2284  2297  2310  2323  2336  2349  2362 

2375  2388  2401  2414  2427  2440  2453  2466  2479  2492 

2504  2517  2530  2543  2556  2569  2582  2595  2608  2621 

2634  2647  2660  2673  2686  2699  2711  2724  2737  2750 

2763  2776  2789  2802  2815  2827  2840  2853  2866  2879 

2892  2905  2917  2930  2943  2956  2969  2982  2994  3007 

3020  3033  3046  3058  3071  3084  3097  3110  3122  3135 

53148  53161  53173  53186  53199  53212  53224  53237  53250  53263 

3275  §288  3301  3314  3326  3339  3352  3364  3377  3390 

3403  3415  3428  3441  3453  3466  3479  3491  3504  3517 

3529  3542  3555  3567  3580  3593  3605  3618  3631  3643 

3656  3668  3681  3694  3706  3719  3732  3744  3757  3769 

3782  3794  3807  3820  3832  3845  3857  3870  3882  3895 

3908  3920  3933  3945  3958  3970  3983  3995  4008  4020 

4033  4045  4058  4070  4083  4095  4108  4120  4133  4145 

4158  4170  4183  4195  4208  4220  4233  4245  4258  4270 

4283  4295  4307  4320  4332  4345  4357  4370  4382  4394 

54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 


TABLE  VI.— LOGARITHMS  OF  NUMBERS.  225 


N        0123456        7        89 


350 

1 
2 
3 

4 
5 
6 
7 
8 
9 

360 

1 


4 
5 
6 
7 
8 
9 

370 

1 
2 
3 
4 
5 
6 
7 


380 

1 
2 
3 

4 
5 
6 

7 


390 
1 
2 
3 
4 
5 


54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 

4531  4543  4555  4568  4580  4593  4605  4617  4630  4642 

4654  4667  4679  4691  4704  4716  4728  4741  4753  4765 

4777  4790  4802  4814  4827  4839  4851  4864  4876  4888 

4900  4913  4925  4937  4949  4962  4974  4986  4998  5011 

5023  5035  5047  5060  5072  5084  5096  5108  5121  5133 

5145  5157  5169  5182  5194  5206  5218  5230  5242  5255 

5267  5279  5291  5303  5315  5328  5340  5352  5364  5376 

5400  5413  5425  5437  5449  5461  5473  5485  5497 

5509  5522  5534  5546  5558  5570  5582  5594  5606  5618 

55630  55642  55654  55666  55678  55691  55703  55715  55727  55739 

5751  5763  5775  5787  5799  5811  5823  5835  5847  5859 

5871  5883  5895  5907  5919  5931  5943  5955  5967  5979 

5991  6003  6015  6027  6038  6050  6062  6074  6086 


6110  6122  6134  6146  6158  6170  6182  6194  6205  6217 

6229  6241  6253  6265  6277  6289  6301  6312  6324  6336 

6348  6360  6372  6384  6396  6407  6419  6431  6443  6455 

6467  6478  6490  6502  6514  6526  6538  6549  6561  6573 

6585  6597  6608  6620  6632  6644  6656  6667  6679  6691 

6703  6714  6726  6738  6750  6761  6773  6785  6797  6808 

56820  56832  56844  56855  56867  56879  56891  56902  56914  56926 

6937  6949  6961  6972  6984  6996  7008  7019  7031  7043 

7054  7066  7078  7089  7101  7113  7124  7136  7148  7159 

7171  7183  7194  7206  7217  7229  7241  7252  7264  7276 

7287  7299  7310  7322  7334  7345  7357  7368  7380  7392 

7403  7415  7426  7438  7449  7461  7473  7484  7496  7507 

7519  7530  7542  7553  7565  7576  7588  7600  7611  7623 

7634  7646  7657  7669  7680  7692  7703  7715  7726  7738 

7749  7761  7772  7784  7795  7807  7818  7830  7841  7852 

7864  7875  7887  7898  7910  7921  7933  7944  7955  7967 

57978  57990  58001  58013  58024  58035  58047  58058  58070  58081 

8092  8104  8115  8127  8138  8149  8161  8172  8184  8195 

8206  8218  8229  8240  8252  8263  8274  8286  8297  8309 

8320  8331  8343  8354  8365  8377  8388  8399  8410  8422 

8433  8444  8456  8467  8478  8490  8501  8512  8524  8535 

8546  8557  8569  8580  8591  8602  8614  8625  8636  8647 

8659  8670  8681  8692  8704  8715  8726  8737  8749  8760 

8771  8782  8794  8805  8816  8827  8838  8850  8861  8872 

8883  8894  8906  8917  8928  8939  8950  8961  8973  8984 

8995  9006  9017  9028  9040  9051  9062  9073  9084  9095 

59106  59118  59129  59140  59151  59162  59173  59184  59195  59207 

9218  9229  9240  9251  9262  9273  9284  9295  9306  931 

9329  9340  9351  9362  9373  9384  9395  9406  9417  94 

9439  9450  9461  9472  9483  9494  9506  9517  9528  9 

9550  9561  9572  9583  9594  9605  9616  9627  9638  964^ 

9660  9671  9682  9693  9704  9715  9726  9737  9748  9759 

9770  9780  9791  9802  9813  9824  9835  9846  9857  9868 

9879  9890  9901  9912  9923  9934  9945  9956  9966  9977 

9988  9999  60010  60021  60032  60043  60054  60065  60076  60086 

9  60097  60108  0119  0130  0141  0152  0163  0173  0184  0195 

400  60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 


226  TABLE  V1.—LOGAKITHMS  OP  NUMBERS. 


NO1234567-89 


400 

1 
2 
3 
4 
5 
6 
7 
8 
9 

410 

1 
2 
3 
4 
5 
6 
7 
8 
9 

420 

1 
2 
3 
4 
5 
6 
7 
8 
9 

430 

1 
2 
3 
4 


7 
8 
9 

440 

1 
2 
3 
4 
5 


450 


60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 

0314  0325  0336  0347  0358  0369  0379  0390  0401  0412 

0423  0433  0444  0455  0466  0477  0487  0498  0509  0520 

0531  0541  0552  0563  0574  0584  0595  0606  0617  0627 

0638  0649  0660  0670  0681  0692  0703  0713  0724  0735 

0746  0756  0767  0778  0788  0799  0810  0821  0831  0842 

0853  0863  0874  0885  0895  0906  0917  0927  0938  0949 

0959  0970  0981  0991  1002  1013  1023  1034  1045  1055 

1066  1077  1087  1098  1109  1119  1130  1140  1151  1162 

1172  1183  1194  1204  1215  1225  1236  1247  1257  1268 

61278  61289  61300  61310  61321  61331  61342  61352  61363  61374 

1384  1395  1405  1416  1426  1437  1448  1458  1469  1479 

1490  1500  1511  1521  1532  1542  1553  1563  1574  1584 

1595  1606  1616  1627  1637  1648  1658  1669  1679  1690 

1700  1711  1721  1731  1742  1752  1763  1773  1784  1794 

1805  1815  1826  1836  1847  1857  1868  1878  1888  1899 

1909  1920  1930  1941  1951  1962  1972  1982  1993  2003 

2014  2024  2034  2045  2055  2066  2076  2086  2097  2107 

2118  2128  2138  2149  2159  2170  2180  2190  2201  2211 

2221  2232  2242  2252  2263  2273  2284  2294  2304  2315 

62325  62335  62346  62356  62366  62377  62387  62397  62408  62418 

2428  2439  2449  2459  2469  2480  2490  2500  2511  2521 

2531  2542  2552  2562  2572  2583  2593  2603  2613  2624 

2634  2644  2655  2665  2675  2685  2696  2706  2716  2726 

2737  2747  2757  2767  2778  2788  2798  2808  2818  2829 

2839  2849  2859  2870  2880  2890  2900  2910  2921  2931 

2941  2951  2961  2972  2982  2992  3002  3012  3022  3033 

3043  3053  3063  3073  3083  3094  3104  3114  3124  3134 

3144  3155  3165  3175  3185  3195  3205  3215  3225  3236 

3246  3256  3266  3276  3286  3296  3306  3317  3327  3337 

63347  63357  63367  63377  63387  63397  63407  63417  63428  63438 

3448  3458  3468  3478  3488  3498  3508  3518  3528  3538 

3548  3558  3568  3579  3589  3599  3609  3619  3629  3639 

3649  3659  3669  3679  3689  3699  3709  3719  3729  3739 

3749  3759  3769  3779  3789  3799  3809  3819  3829  3839 

3849  3859  3869  3879  3889  3899  3909  3919  3929  3939 

3949  3959  3969  3979  3988  3998  4008  4018  4028  4038 

4048  4058  4068  4078  4088  4098  4108  4118  4128  4137 

4147  4157  4167  4177  4187  4197  4207  4217  4227  4237 

4246  4256  4266  4276  4286  4296  4306  4316  4326  4335 

64345  64355  64365  64375  64385  64395  64404  64414  64424  64434 

4444  4454  4464  4473  4483  4493  4503  4513  4523  4532 

4542  4552  4562  4572  4582  4591  4601  4611  4621  4631 

4640  4650  4660  4670  4680  4689  4699  4709  4719  4729 

4738  4748  4758  4768  4777  4787  4797  4807  4816  4826 

4836  4846  4856  4865  4875  4885  4895  4904  4914  4924 

4933  4943  4953  4963  4972  4982  4992  5002  5011  5021 

5031  5040  5050  5060  5070  5079  5089  5099  5108  5118 

5128  5137  5147  5157  5167  5176  5186  5196  5205  5215 

5225  6234  5244  5254  6263  5273  5283  5292  5302  5312 

65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 


TABLE  VI.— LOGARITHMS  OF  NUMBEUS.  227 


N 


0123456789 


450 

1 
2 
3 

4 
5 
6 
7 
8 


460 

1 
2 
3 
4 
5 
6 
7 
8 
9 

470 

1 
2 
3 

4 
5 
6 
7 
8 
9 

480 

1 
2 
3 

4 
5 
6 
7 
8 
9 

490 
1 
2 
3 
4 
5 
6 
7 


500 


65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 

5418  5427  5437  5447  5456  5466  5475  5485  5495  5504 

5514  6523  5533  5543  5552  5562  5571  5581  5591  5600 

5610  5619  5629  5639  5648  5658  5667  5677  5686  5696 

5706  5715  5725  5734  5744  5753  5763  5772  5782  5792 

5801  5811  5820  5830  5839  5849  5858  5868  5877  5887 

5896  5906  5916  5925  5935  5944  5954  5963  5973  5982 

5992  6001  6011  6020  6030  6039  6049  6058  6068  6077 

6087  6096  6106  6115  6124  6134  6143  6153  6162  6172 

6181  6191  6200  6210  6219  6229  6238  6247  6257  6266 

66276  66285  66295  66304  66314  66323  66332  66342  66351  66361 

6370  6380  6389  6398  6408  6417  6427  6436  6445  6455 

6464  6474  6483  6492  6502  6511  6521  6530  6539  6549 

6558  6567  6577  6586  6596  6605  6614  6624  6633  6642 

6652  6661  6671  6680  6689  6699  6708  6717  6727  6736 

6745  6755  6764  6773  6783  6792  6801  6811  6820  6829 

6839  6848  6857  6867  6876  6885  6894  6904  6913  6922 

6932  6941  6950  6960  6969  6978  6987  6997  7006  7015 

7025  7034  7043  7052  7062  7071  7080  7089  7099  7108 

7117  7127  7136  7145  7154  7164  7173  7182  7191  7201 

67210  67219  67228  67237  67247  67256  67265  67274  67284  67293 

7302  7311  7321  7330  7339  7348  7357  7367  7376  7385 

7394  7403  7413  7422  7431  7440  7449  7459  7468  7477 

7486  7495  7504  7514  7523  7532  7541  7550  7560  7569 

7578  7587  7596  7605  7614  7624  7633  7642  7651  7660 

7669  7679  7688  7697  7706  7715  7724  7733  7742  7752 

7761  7770  7779  7788  7797  7806  7815  7825  7834  7843 

7852  7861  7870  7879  7888  7897  7906  7916  7925  7934 

7943  7952  7961  7970  7979  7988  7997  8006  8015  8024 

8034  8043  8052  8061  8070  8079  8088  8097  8106  8115 

68124  68133  68142  68151  68160  68169  68178  68187  68196  68205 

8215  8224  8233  8242  8251  8260  8269  8278  8287  8296 

8305  8314  8323  8332  8341  8350  8359  8368  8377  8386 

8395  8404  8413  8422  8431  8440  8449  8458  8467  8476 

8485  8494  8502  8511  8520  8529  8538  8547  8556  8565 

8574  8583  8592  8601  8610  8619  8628  8637  8646  8655 

8664  8673  8681  8690  8699  8708  8717  8726  8735  8744 

8753  8762  8771  8780  8789  8797  8806  8815  8824  8833 

8842  8851  8860  8869  8878  8886  8895  8904  8913  8922 

8931  8940  8949  8958  8966  8975  8984  8993  9002  9011 

69020  69028  69037  69046  69055  69064  69073  69082  69090  69099 

9108  9117  9126  9135  9144  9152  9161  9170  9179  9188 

9197  9205  9214  9223  9232  9241  9249  9258  9267  9276 

9285  9294  9302  9311  9320  9329  9338  9346  9355  9364 

9373  9381  9390  9399  9408  9417  9425  9434  9443  9452 

9461  9469  9478  9487  9496  9504  9513  9522  9531  9539 

9548  9557  9566  9574  9583  9592  9601  9609  9618  9627 

9636  9644  9653  9662  9671  9679  9688  9697  9705  9714 

9723  9732  9740  9749  9758  9767  9775  9784  9793  9801 

9810  9819  9827  9836  9845  9854  9862  9871  9880 

69897  69906  69914  69923  69932  69940  69949  69958  69966  69975 


TABLE  VI.—  LOGARITHMS  OF  NUMBERS. 


N012345O789 


500 

1 
2 
3 
4 
5 
6 
7 
8 


510 

1 

2 
3 
4 
5 


520 

1 
2 
3 
4 
5 
6 
7 
8 
9 

580 

1 
2 
3 
4 
5 
6 
7 


540 

1 
2 
3 

4 
5 
6 

7 


550 


(50807  69906  69914  69923  69932  69940  69949  69958  69966  (59975 

9984  9992  70001  70010  70018  70027  70036  70044  70053  70062 

70070  70079  0088  0096  0105  0114  0122  0131  0140  0148 

0157  0165  0174  0183  0191  0200  0209  0217  0226  0234 

0243  0252  0260  0269  0278  0286  0295  0303  0312  0321 

0329  0338  0346  0355_  0364  0372  0381  0389  0398  0406 

0415  0424  0432  0441  0449  0458  0467  0475  0484  0492 

0501  0509  0518  0526  0535  0544  0552  0561  0569  0578 

0586  0595  0603  0612  0621  0629  0638  0646  0655  0663 

0672  0680  0689  0697  0706  0714  0723  0731  0740  0740 

70757  70766  70774  70783  70791  70800  70808  70817  70825  70834 

0842  0851  0859  0868  0876  0885  0893  0902  0910  0919 

0927  0935  0944  0952  0961  0969  0978  0986  0995  1003 

1012  1020  1029  1037  1046  1054  1063  1071  1079  1088 

1096  1105  1113  1122  1130  1139  1147  1155  1164  1172 

1181  1189  1198  1206  1214  1223  1231  1240  1248  1257 

1265  1273  1282  1290  1299  1307  1315  1324  1332  1341 

1349  1357  1366  1374  1383  1391  1399  1408  1416  1425 

1433  1441  1450  1458  1466  1475  1483  1492  1500  1508 

1517  1525  1533  1542  1550  1559  1567  1575  1584  1592 

71600  71609  71617  71625  71634  71642  71650  71659  71667  71675 

1684  1692  1700  1709  1717  1725.1734  1742  1750  1750 

1767  1775  1784  1792  1800  1809  1817  1825  1834  1842 

1850  1858  1867  1875  1883  1892  1900  1908  1917  1925 

1933  1941  1950  1958  1966  1975  1983  1991  1999  2008 

2016  2024  2032  2041  2049  2057  2066  2074  2082  2090 

2099  2107  2115  2123  2132  2140  2148  2156  2165  2173 

2181  2189  2198  2206  2214  2222  2230  2239  2247  2255 

2263  2272  2280  2288  2296  2304  2313  2321  2329  2337 

2346  2354  2362  2370  2378  2387  2395  2403  2411  2419 

72428  72436  72444  72452  72460  72469  72477  72485  72493  72501 

2509  2518  2526  2534  2542  2550  2558  2567  2575  2583 

2591  2599  2607  2616  2624  2632  2640  2648  2656  2665 

2673  2681  2689  2697  2705  2713  2722  2730  2738  2746 

2754  2762  2770  2779  2787  2795  2803  2811  2819  2827 

2835  2843  2852  2860  2868  2876  2884  2892  2900  2908 

2916  2925  2933  2941  2949  2957  2965  2973  2981  2989 

2997  3006  3014  3022  3030  3038  3046  3054  3062  3070 

3078  3086  3094  3102  3111  3119  3127  3135  3143  3151 

3159  3167  3175  3183  3191  3199  3207  3215  3223  3231 

73239  73247  73255  73263  73272  73280  73288  73296  73304  73312 

3320  3328  3336  3344  3352  3360  33(58  3376  3384  3392 

3400  3408  3416  3424  3432  3440  3448  3456  3464  3472 

3480  3488  3496  3504  3512  3520  3528  3536  3544  3552 

3560  3568  3576  3584  3592  3600  3608  361(5  .'5(',24  3032 

3640  3648  3656  3664  3672  3679  3687  3695  3703  3711 

3719  3727  3735  3743  3751  3759  3767  3775  3783  3701 

3799  3807  3815  3823  3830  3838  3846  3854  3862  3870 

3878  3886  3894  3902  3910  3918  3926  3933  3941  ;',040 

3957  3965  3973  3981  3989  3997  4005  4013  4020  4028 

74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 


TABLE  VI.—  LOGARITHMS  OF  NUMBERS.     229 


6789 


550 

1 
2 
3 
4 
5 
6 
7 
8 
9 

560 
1 

2 
3 
4 
5 
6 
7 
8 
9 

570 

1 

2 
3 
4 
5 
6 
7 
8 
9 

580 
1 
2 
3 
4 
5 
6 
7 
8 
9 

590 
1 
2 
3 
4 
5 
6 
7 
8 
9 

600 


74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 

4H5  4123  4131  4139  4147  4155  4162  4170  4178  4186 

4194  4202  4210  4218  4225  4233  4241  4249  4257  4265 

4273  4280  4288  4296  4304  4312  4320  4327  4335  4343 

4351  4359  4367  4374  4382  4390  4398  4406  4414  4421 

4429  4437  4445  4453  4461  4468  4476  4484  4492  4500 

4507  4515  4523  4531  4539  4547  4554  4562  4570  4578 

4586  4593  4601  4609  4617  4624  4632  4640  4648  4656 

4663  4671  4679  4687  4695  4702  4710  4718  4726  4733 

4741  4749  4757  4764  4772  4780  4788  4796  4803  4811 

74819  74827  74834  74842  74850  74858  74865  74873  74881  74889 

4896  4904  4912  4920  4927  4935  4943  4950  4958  4966 

4974  4981  4989  4997  5005  5012  5020  5028  5035  5043 

5051  5059  5066  5074  5082  5089  5097  5105  5113  5120 

5128  5136  5143  5151  5159  5166  5174  5182  5189  5197 

5205  5213  5220  5228  5236  5243  5251  5259  5266  5274 

5282  5289  5297  5305  5312  5320  5328  5335  5343  5351 

5358  5366  5374  5381  5389  5397  5404  5412  5420  5427 

5435  5442  5450  5458  5465  5473  5481  5488  5496  5504 

5511  5519  5526  5534  5542  5549  5557  5565  5572  5580 

75587  75595  75603  75610  75618  75626  75633  75641  75648  75656 

5664  5671  5679  5686  5694  5702  5709  5717  5724  5732 

5740  5747  5755  5762  5770  5778  5785  5793  5800  5808 

5815  5823  5831  5838  5846  5853  5861  5868  5876  5884 

5891  5899  5906  5914  5921  5929  5937  5944  5952  5959 

5967  5974  5982  5989  5997  6005  6012  6020  6027  6035 

6042  6050  6057  6065  6072  6080  6087  6095  6103  6110 

6118  6125  6133  6140  6148  6155  6163  6170  6178  6185 

6193  6200  6208  6215  6223  6230  6238  6245  6253  6260 

6268  6275  6283  6290  6298  6305  6313  6320  6328  6335 

76343  76350  76358  76365  76373  76380  76388  76395  76403  76410 

6418  6425  6433  6440  6448  6455  6462  6470  6477  6485 

6492  6500  6507  6515  6522  6530  6537  6545  6552  6559 

6567  6574  6582  6589  6597  6604  6612  6619  6626  6634 

6641  6649  6656  6664  6671  6678  6686  6693  6701  6708 

6716  6723  6730  6738  6745  6753  6760  6768  6775  6782 

6790  6797  6805  6812  6819  6827  6834  6842  6849  6856 

6864  6871  6879  6886  6893  6901  6908  6916  6923  6930 

6938  6945  6953  6960  6967  6975  6982  6989  6997  7004 

7012  7019  7026  7034  7041  7048  7056  7063  7070  7078 

77085  77093  77100  77107  77115  77122  77129  77137  77144  77151 

7159  7166  7173  7181  7188  7195  7203  7210  7217  7225 

7232  7240  7247  7254  7262  7269  7276  7283  7291  7298 

7305  7313  7320  7327  7335  7342  7349  7357  7364  7371 

7379  7386  7393  7401  7408  7415  7422  7430  7437  7444 

7452  7459  7466  7474  7481  7488  7495  7503  7510  7517 

7525  7532  7539  7546  7554  7561  7568  7576  7583  7590 

7597  7605  7612  7619  7627  7634  7641  7648  7656  7663 

7670  7677  7685  7692  7699  7706  7714  7721  7728  7735 

7743  7750  7757  7764  7772  7779  7786  7793  7801  7808 

77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 


230  TABLE  VI.—  LOGARITHMS   OF  NUMBERS. 


600 
1 

2 
3 
4 
5 
6 
7 
8 
9 

610 
1 
2 
3 

4 
5 
6 
7 
8 
9 

620 
1 
2 
3 

4 
5 
6 
7 
8 
9 

630 
1 

2 
3 

4 


7 
8 
9 

640 
1 
2 
3 

4 
5 
6 
7 
8 
9 

650 


23456789 


77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 

7887  7895  7902  7909  7916  7924  7931  7938  7945  7952 

7960  7967  7974  7981  7988  7996  8003  8010  8017  8025 

8032  8039  8046  8053  8061  8068  8075  8082  8089  8097 

8104  8111  8118  8125  8132  8140  8147  8154  8161  8168 

8176  8183  8190  8197  8204  8211  8219  8226  8233  8240 

8247  8254  8262  8269  8276  8283  8290  8297  8305  8312 

8319  8326  8333  8340  8347  8355  8362  8369  8376  8383 

8390  8398  8405  8412  8419  8426  8433  8440  8447  8455 

8462  8469  8476  8483  8490  8497  8504  8512  8519  8526 

78533  78540  78547  78554  78561  78569  78576  78583  78590  78597 

8604  8611  8618  8625  8633  8640  8647  8654  8661  8668 

8675  8682  8689  8696  8704  8711  8718  8725  8732  8739 

8746  8753  8760  8767  8774  8781  8789  8796  8803  8810 

8817  8824  8831  8838  8845  8852  8859  8866  8873  8880 

8888  8895  8902  8909  8916  8923  8930  8937  8944  8951 

8958  8965  8972  8979  8986  8993  9000  9007  9014  9021 

9029  9036  9043  9050  9057  9064  9071  9078  9085  9092 

9099  9106  9113  9120  9127  9134  9141  9148  9155  9162 

9169  9176  9183  9190  9197  9204  9211  9218  9225  9232 

79239  79246  79253  79260  79267  79274  79281  79288  79295  79302 

9309  9316  9323  9330  9337  9344  9351  9358  9365  9372 

9379  9386  9393  9400  9407  9414  9421  9428  9435  9442 

9449  9456  9463  9470  9477  9484  9491  9498  9505  9511 

9518  9525  9532  9539  9546  9553  9560  9567  9574  9581 

9588  9595  9602  9609  9616  9623  9630  9637  9644  9650 

9657  9664  9671  9678  9685  9692  9699  9706  9713  9720 

9727  9734  9741  9748  9754  9761  9768  9775  9782  9789 

9796  9803  9810  9817  9824  9831  9837  9844  9851  9858 

9865  9872  9879  9886  9893  9900  9906  9913  9920  9927 

79934  79941  79948  79955  79962  79969  79975  79982  79989  79996 

80003  80010  80017  80024  80030  80037  80044  80051  80058  80065 

0072  0079  0085  0092  0099  0106  0113  0120  0127  0134 

0140  0147  0154  0161  0168  0175  0182  0188  0195  0202 

0209  0216  0223  0229  0236  0243  0250  0257  0264  0271 

0277  0284  0291  0298  0305  0312  0318  0325  0332  0339 

0346  0353  0359  0366  0373  0380  0387  0393  0400  0407 

0414  0421  0428  0434  0441  0448  0455  0462  0468  0475 

0482  0489  0496  0502  0509  0516  0523  0530  0536  0543 

0550  0557  0564  0570  0577  0584  0591  0598  0604  0611 

80618  80625  80632  80638  80645  80652  80659  80665  80672  80679 

0686  0693  0699  0706  0713  0720  0726  0733  0740  0747 

0754  0760  0767  0774  0781  0787  0794  0801  0808  0814 

0821  0828  0835  0841  0848  0855  0862  0868  0875  0882 

0889  0895  0902  0909  0916  0922  0929  0936  0943  0949 

0956  0963  0969  0976  0983  0990  0996  1003  1010  1017 

1023  1030  1037  1043  1050  1057  1064  1070  1077  1084 

1090  1097  1104  1111  1117  1124  1131  1137  1144  1151 

1158  1164  1171  1178  1184  1191  1198  1204  1211  1218 

1224  1231  1238  1245  1251  1258  1265  1271  1278  1285 

81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 


TABLE  VI.— LOGARITHMS  OF  NUMB^m*.     231 


8   9 


650 

1 
2 
3 

4 
5 
6 
7 
8 
9 

660 

1 
2 
3 
4 
5 
6 
7 
8 
9 

670 

1 
2 
3 

4 
5 
6 
7 
8 
9 

680 
1 
2 
3 

4 
5 
6 
7 
8 
9 

690 
1 
2 
3 
4 
5 
6 
7 


700 


81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

1358  1365  1371  1378  1385  1391  1398  1405  1411  1418 

1425  1431  1438  1445  1451  1458  1465  1471  1478  1485 

1491  1498  1505  1511  1518  1525  1531  1538  1544  1551 

1558  1564  1571  1578  1584  1591  1598  1604  1611  1617 

1624  1631  1637  1644  1651  1657  1664  1671  1677  1684 

1690  1697  1704  1710  1717  1723  1730  1737  1743  1750 

1757  1763  1770  1776  1783  1790  1796  1803  1809  1816 

1823  1829  1836  1842  1849  1856  1862  1869  1875  1882 

1889  1895  1902  1908  1915  1921  1928  1935  1941  1948 

81954  81961  81968  81974  81981  81987  81994  82000  82007  82014 

2020  2027  2033  2040  2046  2053  2060  2066  2073  2079 

2086  2092  2099  2105  2112  2119  2125  2132  2138  2145 

2151  2158  2164  2171  2178  2184  2191  2197  2204  2210 

2217  2223  2230  2236  2243  2249  2256  2263  2269  2276 

2282  2289  2295  2302  2308  2315  2321  2328  2334  2341 

2347  2354  2360  2367  2373  2380  2387  2393  2400  2406 

2413  2419  2426  2432  2439  2445  2452  2458  2465  2471 

2478  2484  2491  2497  2504  2510  2517  2523  2530  2536 

2543  2549  2556  2562  2569  2575  2582  2588  2595  2601 

82607  82614  82620  82627  82633  82640  82646  82653  82659  82666 

2672  2679  2685  2692  2698  2705  2711  2718  2724  2730 

2737  2743  2750  2756  2763  2769  2776  2782  2789  2795 

2802  2808  2814  2821  2827  2834  2840  2847  2853  2860 

2866  2872  2879  2885  2892  2898  2905  2911  2918  2924 

2930  2937  2943  2950  2956  2963  2969  2975  2982  2988 

2995  3001  3008  3014  3020  3027  3033  3040  3046  3052 

3059  3065  3072  3078  3085  3091  3097  3104  3110  3117 

3123  3129  3136  3142  3149  3155  3161  3168  3174  3181 

3187  3193  3200  3206  3213  3219  3225  3232  3238  3245 

83251  83257  83264  83270  83276  83283  83289  83296  83302  83308 

3315  3321  3327  3334  3340  3347  3353  3359  3366  3372 

3378  3385  3391  3398  3404  3410  3417  3423  3429  3436 

3442  3448  3455  3461  3467  3474  3480  3487  3493  3499 

3506  3512  3518  3525  3531  3537  3544  3550  3556  3563 

3569  3575  3582  3588  3594  3601  3607  3613  3620  3626 

3632  3639  3645  3651  3658  3664  3670  3677  3683  3689 

3696  3702  3708  3715  3721  3727  3734  3740  3746  3753 

3759  3765  3771  3778  3784  3790  3797  3803  3809  3816 

3822  3828  3835  3841  3847  3853  3860  3866  3872  3879 

83885  83891  83897  83904  83910  83916  83923  83929  83935  83942 

3948  3954  3960  3967  3973  3979  3985  3992  3998  4004 

4011  4017  4023  4029  4036  4042  4048  4055  4061  4067 

4073  4080  4086  4092  4098  4105  4111  4117  4123  4130 

4136  4142  4148  4155  4161  4167  4173  4180  4186  4192 

4198  4205  4211  4217  4223  4230  4236  4242  4248  4255 

4261  4267  4273  4280  4286  4292  4298  4305  4311  4317 

4323  4330  4336  4342  4348  4354  4361  4367  4373  4379 

4386  4392  4398  4404  4410  4417  4423  4429  4435  4442 

4448  4454  4460  4466  4473  4479  4485  4491  4497  4504 

84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 


232  TABLE  VI.— LOGARITHMS  OF  NUMBERS. 


NO123456789 


700 

1 
2 
3 

4 
5 
6 
7 
8 
9 

710 

1 
2 
3 

4 
5 
6 

7 
8 


720 

1 
2 
3 
4 
5 
6 
7 
8 
9 

730 

1 


740 

1 
2 
3 
4 
5 


750 


84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 
4572  4578  4584  4590  4597  4603  4609  4615  4621  4628 

4634  4640  4646  4652  4658  4665  4671  4677  4683  4689 

4696  4702  4708  4714  4720  4726  4733  4739  4745  4751 

4757  4763  4770  4776  4782  4788  4794  4800  4807  4813 

4819  4825  4831  4837  4844  4850  4856  4862  4868  4874 

4880  4887  4893  4899  4905  4911  4917  4924  4930  4936 

4942  4948  4954  4960  4967  4973  4979  4985  4991  4997 

5003  5009  5016  5022  5028  5034  5040  5046  5052  5058 

5065  5071  5077  5083  5089  5095  5101  5107  5114  5120 

85126  85132  85138  85144  85150  85156  85163  85169  85175  85181 

5187  5193  5199  5205  5211  5217  5224  5230  5236  5242 

5248  5254  5260  5266  5272  5278  5285  5291  5297  5303 

5309  5315  5321  5327  5333  5339  5345  5352  5358  5364 

5370  5376  5382  5388  5394  5400  5406  5412  5418  5425 

5431  5437  5443  5449  5455  5461  5467  5473  5479  5485 

5491  5497  5503  5509  5516  5522  5528  5534  5540  5546 

5552  5558  5564  5570  5576  5582  5588  5594  5600  5606 

5612  5618  5625  5631  5637  5643  5649  5655  5661  5667 

5673  5679  5685  5691  5697  5703  5709  5715  5721  5727 

85733  85739  85745  85751  85757  85763  85769  85775  85781  85788 

5794  5800  5806  5812  5818  5824  5830  5836  5842  5848 

5854  5860  5866  5872  5878  5884  5890  5896  5902  5908 

5914  5920  5926  5932  5938  5944  5950  5956  5962  5968 

5974  5980  5986  5992  5998  6004  6010  6016  6022  6028 

6034  6040  6046  6052  6058  6064  6070  6076  6082  6088 

6094  6100  6106  6112  6118  6124  6130  6136  6141  6147 

6153  6159  6165  6171  6177  6183  6189  6195  6201  6207 

6213  6219  6225  6231  6237  6243  6249  6255  6261  6267 

6273  6279  6285  6291  6297  6303  6308  6314  6320  6326 

86332  86338  86344  86350  86356  86362  86368  86374  86380  86386 

6392  6398  6404  6410  6415  6421  6427  6433  6439  6445 

6451  6457  6463  6469  6475  6481  6487  6493  6499  6504 

6510  6516  6522  6528  6534  6540  6546  6552  6558  6564 

6570  6576  6581  6587  6593  6599  6605  6611  6617  6623 

6629  6635  6641  6646  6652  6658  6664  6670  6676  6682 

6688  6694  6700  6705  6711  6717  6723  6729  6735  6741 

6747  6753  6759  6764  6770  6776  6782  6788  6794  6800 

6806  6812  6817  6823  6829  6835  6841  6847  6853  6859 

6864  6870  6876  6882  6888  6894  6900  6906  6911  6917 

86923  86929  86935  86941  86947  86953  86958  86964  86970  86976 

6982  6988  6994  6999  7005  7011  7017  7023  7029  7035 

7040  7046  7052  7058  7064  7070  7075  7081  7087  7093 

7099  7105  7111  7116  7122  7128  7134  7140  7146  7151 

7157  7163  7169  7175  7181  7186  7192  7198  7204  7210 

7216  7221  7227  7233  7239  7245  7251  7256  7262  7268 

7274  7280  7286  7291  7297  7303  7309  7315  7320  7326 

7332  7338  7344  7349  7355  7361  7367  7373  7379  7384 

7390  7396  7402  7408  7413  7419  7425  7431  7437  7442 

7448  7454  7460  7466  7471  7477  7483  7489  7495  7500 

87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 


TABLE  VI.— LOGARITHMS   OF   NUMBERS.  233 


750 

1 
2 
3 

4 
5 
6 
7 
8 
9 

760 

1 
2 
3 

4 
5 
6 
7 
8 
9 

770 

1 
2 
3 

4 
5 
6 

7 
8 
9 

780 
1 
2 
3 
4 
5 
6 
7 
8 
9 

790 

1 
2 
3 
4 
5 
6 
7 
8 
9 

800 


345678 


87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 

7564  7570  7576  7581  7587  7593  7599  7604  7610  7616 

7622  7628  7633  7639  7645  7651  7656  7662  7668  7674 

7679  7685  7691  7697  7703  7708  7714  7720  7726  7731 

7737  7743  7749  7754  7760  7766  7772  7777  7783  7789 

7795  7800  7806  7812  7818  7823  7829  7835  7841  7846 

7852  7858  7864  7869  7875  7881  7887  7892  7898  7904 

7910  7915  7921  7927  7933  7938  7944  7950  7955  7961 

7967  7973  7978  7984  7990  7996  8001  8007  8013  8018 

8024  8030  8036  8041  8047  8053  8058  8064  8070  8076 

88081  88087  88093  88098  88104  88110  88116  88121  88127  88133 

8138  8144  8150  8156  8161  8167  8173  8178  8184  8190 

8195  8201  8207  8213  8218  8224  8230  8235  8241  8247 

8252  8258  8264  8270  8275  8281  8287  8292  8298  8304 

8309  8315  8321  8326  8332  8338  8343  8349  8355  8360 

8366  8372  8377  8383  8389  8395  8400  8406  8412  8417 

8423  8429  8434  8440  8446  8451  8457  8463  8468  8474 

8480  8485  8491  8497  8502  8508  8513  8519  8525  8530 

8536  8542  8547  8553  8559  8564  8570  8576  8581  8587 

8593  8598  8604  8610  8615  8621  8627  8632  8638  8643 

88649  88655  88660  88666  88672  88677  88683  88689  88694  88700 

8705  8711  8717  8722  8728  8734  8739  8745  8750  8756 

8762  8767  8773  8779  8784  8790  8795  8801  8807  8812 

8818  8824  8829  8835  8840  8846  8852  8857  8863  8868 

8874  8880  8885  8891  8897  8902  8908  8913  8919  8925 

8930  8936  8941  8947  8953  8958  8964  8969  8975  8981 

8986  8992  8997  9003  9009  9014  9020  9025  9031  9037 

9042  9048  9053  9059  9064  9070  9076  9081  9087  9092 

9098  9104  9109  9115  9120  9126  9131  9137  9143  9148 

9154  9159  9165  9170  9176  9182  9187  9193  9198  9204 

89209  89215  89221  89226  89232  89237  89243  89248  89254  89260 

9265  9271  9276  9282  9287  9293  9298  9304  9310  9315 

9321  9326  9332  9337  9343  9348  9354  9360  9365  9371 

9376  9382  9387  9393  9398  9404  9409  9415  9421  9426 

9432  9437  9443  9448  9454  9459  9465  9470  9476  9481 

9487  9492  9498  9504  9509  9515  9520  9526  9531  9537 

9542  9548  9553  9559  9564  9570  9575  9581  9586  9592 

9597  9603  9609  9614  9620  9625  9631  9636  9642  9647 

9653  9658  9664  9669  9675  9680  9686  9691  9697  9702 

9708  9713  9719  9724  9730  9735  9741  9746  9752  9757 

89763  89768  89774  89779  89785  89790  89796  89801  89807  89812 

9818  9823  9829  9834  9840  9845  9851  9856  9862  9867 

9873  9878  9883  9889  9894  9900  9905  9911  9916  9922 

9927  9933  9938  9944  9949  9955  9960  9966  9971  9977 

9982  9988  9993  9998  90004  90009  90015  90020  90026  90031 

90037  90042  90048  90053  0059  0064  0069  0075  0080  0086 

0091  0097  0102  0108  0113  0119  0124  0129  0135  0140 

0146  0151  0157  0162  0168  0173  0179  0184  0189  0195 

0200  0206  0211  0217  0222  0227  0233  0238  0244  0249 

0255  0260  0266  0271  0276  0282  0287  0293  0298  0304 

90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 


234  TABLE  VI.  —LOGARITHMS   OF   NUMBERS. 


N 


3456789 


800 

1 
2 
3 

4 
5 
6 
7 
8 
9 

810 

1 
2 
3 
4 
5 
6 
7 
8 
9 

820 
1 
2 
3 
4 
5 
6 
7 


830 

1 
2 
3 

4 
5 
6 

7 
8 


840 

1 
2 
3 
4 
5 
6 
7 
8 
9 

850 


90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

0363  0369  0374  0380  0385  0390  0396  0401  0407  0412 

0417  0423  0428  0434  0439  0445  0450  0455  0461  0466 

0472  0477  0482  0488  0493  0499  0504  0509  0515  0520 

0526  0531  0536  0542  0547  0553  0558  0563  0569  0574 

0580  0585  0590  0596  0601  0607  0612  0617  0623  0628 

0634  0639  0644  0650  0655  0660  0666  0671  0677  0682 

0687  0693  0698  0703  0709  0714  0720  0725  0730  0736 

0741  0747  0752  0757  0763  0768  0773  0779  0784  0789 

0795  0800  0806  0811  0816  0822  0827  0832  0838  0843 

90849  90854  90859  90865  90870  90875  90881  90886  90891  90897 

0902  0907  0913  0918  0924  0929  0934  0940  0945  0950 

0956  0961  0966  0972  0977  0982  0988  0993  0998  1004 

1009  1014  1020  1025  1030  1036  1041  1046  1052  1057 

1062  1068  1073  1078  1084  1089  1094  1100  1105  1110 

1116  1121  1126  1132  1137  1142  1148  1153  1158  1164 

1169  1174  1180  1185  1190  1196  1201  1206  1212  1217 

1222  1228  1233  1238  1243  1249  1254  1259  1265  1270 

1275  1281  1286  1291  1297  1302  1307  1312  1318  1323 

1328  1334  1339  1344  1350  1355  1360  1365  1371  1376 

91381  91387  91392  91397  91403  91408  91413  91418  91424  91429 

1434  1440  1445  1450  1455  1461  1466  1471  1477  1482 

1487  1492  1498  1503  1508  1514  1519  1524  1529  1535 

1540  1545  1551  1556  1561  1566  1572  1577  1582  1587 

1593  1598  1603  1609  1614  1619  1624  1630  1635  1640 

1645  1651  1656  1661  1666  1672  1677  1682  1687  1693 

1703  1709  1714  1719  1724  1730  1735  1740  1745 

1751  1756  1761  1766  1772  1777  1782  1787  1793  1798 

1803  1808  1814  1819  1824  1829  1834  1840  1845  1850 

1855  1861  1866  1871  1876  1882  1887  1892  1897  1903 

91908  91913  91918  91924  91929  91934  91939  91944  9195.0  91955 

1960  1965  1971  1976  1981  1986  1991  1997  2002  2007 

2012  2018  2023  2028  2033  2038  2044  2049  2054  2059 

2065_  2070  2075  2080  2085  2091  2096  2101  2106  2111 

2117  2122  2127  2132  2137  2143  2148  2153  2158  2163 

2169  2174  2179  2184  2189  2195  2200  2205  2210  2215 

2221  2226  2231  2236  2241  2247  2252  2257  2262  2267 

2273  2278  2283  2288  2293  2298  2304  2309  2314  2319 

2324  2330  2335  2340  2345  2350  2355  2361  2366  2371 

2376  2381  2387  2392  2397  2402  2407  2412  2418  2423 

92428  92433  92438  92443  92449  92454  92459  92464  92469  92474 

2480  2485  2490  2495  2500  2505  2511  2516  2521  2526 

2531  2536  2542  2547  2552  2557  2562  2567  2572  2578 

2583  2588  2593  2598  2603  2609  2614  2619  2624  2629 

2634  2639  2645  2650  2655  2660  2665  2670  2675  2681 

2686  2691  2696  2701  2706  2711  2716  2722  2727  2732 

2737  2742  2747  2752  2758  2763  2768  2773  2778  2783 

2788  2793  2799  2804  2809  2814  2819  2824  2829  2834 

2840  2845  2850  2855  2860  2865  2870  2875  2881  2886 

2891  2896  2901  2906  2911  2916  2921  2927  2932  2937 

92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 


TABLE  VI.— LOGARITHMS   OF   NUMBERS.  235 


NO123456789 


850 

1 
2 
3 
4 
5 
6 
7 
8 
9 

860 

1 
2 
3 

4 
5 
6 

7 


870 
1 
2 
3 
4 
5 


9 

880 
1 
2 
3 
4 
5 
6 
7 
8 
9 

890 
1 
2 
3 
4 
5 
6 
7 
8 
9 

900 


92942  92947  92952  92957  92902  92907  92973  92978  92983  92988 

2993  2998  3003  3008  3013  3018  3024  3029  3034  3039 

3044  3049  3054  3059  3004  3009  3075  3080  3085  3090 

3095  3100  3105  3110  3115  3120  3125  3131  3130  3141 

3140  3151  3150  3101  3100  3171  3170  3181  3180  3192 

3197  3202  3207  3212  3217  3222  3227  3232  3237  3242 

3247  3252  3258  3203  3208  3273  3278  3283  3288  3293 

3298  3303  3308  3313  3318  3323  3328  3334  3339  3344 

3349  3354  3359  3304  3309  3374  3379  3384  3389  3394 

3399  3404  3409  3414  3420  3425  3430  3435  3440  3445 

93450  93455  93400  93405  93470  93475  93480  93485  93490  93495 

3500  3505  3510  3515  3520  3520  3531  3530  3541  3540 

3551  3550  3501  3500  3571  3570  3581  3580  3591  3590 

3001  3000  3011  3010  3021  3020  3031  3030  3041  3040 

3051  3050  3001  3000  3071  3070  3082  3087  3092  3097 

3702  3707  3712  3717  3722  3727  3732  3737  3742  3747 

3752  3757  3702  3707  3772  3777  3782  3787  3792  3797 

3802  3807  3812  3817  3822  3827  3832  3837  3842  3847 

3852  3857  3802  3807  3872  3877  3882  3887  3892  3897 

3902  3907  3912  3917  3922  3927  3932  3937  3942  3947 

93952  93957  93902  93907  93972  93977  93982  P3987  93992  93997 

4002  4007  4012  4017  4022  4027  4032  4037  4042  4047 

4052  4057  4002  4007  4072  4077  4082  4080  4091  4090 

4101  4100  4111  4110  4121  4120  4131  4130  4141  4140 

4151  4150  4101  4100  4171  4170  4181  4180  4191  4190 

4201  4200  4211  4210  4221  4220  4231  4230  4240  4245 

4250  4255  4200  4205  4270  4275  4280  4285  4290  4295 

4300  4305  4310  4315  4320  4325  4330  4335  4340  4345 

4349  4354  4359  4304  4309  4374  4379  4384  4389  4394 

4399  4404  4409  4414  4419  4424  4429  4433  4438  4443 

94448  94453  94458  94403  94408  94473  94478  94483  94488  94493 

4498  4503  4507  4512  4517  4522  4527  4532  4537  4542 

4547  4552  4557  4502  4507  4571  4570  4581  4580  4591 

4590  4001  4000  4011  4010  4021  4020  4030  4035  4040 

4045  4050  4055  4000  4005  4070  4075  4080  4085  4089 

4094  4099  4704  4709  4714  4719  4724  4729  4734  4738 

4743  4748  4753  4758  4703  4708  4773  4778  4783  4787 

4792  4797  4802  4807  4812  4817  4822  4827  4832  4830 

4841  4840  4851  4850  4801  4800  4871  4870  4880  4885 

4890  4895  4900  4905  4910  4915  4919  4924  4929  4934 

94939  94944  94949  94954  94959  94903  94908  94973  94978  94983 

4988  4993  4998  5002  5007  5012  5017  5022  5027  5032 

5030  5041  5040  5051  5050  5001  5000  5071  5075  5080 

5085  5090  5095  5100  5105  5109  5114  5119  5124  5129 

5134  5139  5143  5148  5153  5158  5103  5108  5173  5177 

5182  5187  5192  5197  5202  5207  5211  5210  5221  5220 

5231  5230  5240  5245  5250  5255  5200  5205  5270  5274 

5279  5284  5289  5294  5299  5303  5308  5313  5318  5323 

5328  5332  5337  5342  5347  5352  5357  5301  5300  5371 

5370  5381  5380  5390  5395  5400  5405  5410  5415  5419 

95424  95429  95434  95439  95444  95448  95453  95458  95403  95408 


236     TABLE  VI.— LOGARITHMS  OF  NUMBERS. 


900 

1 
2 
3 

4 
5 
6 
7 
8 
9 

910 

1 
2 
3 
4 
5 
6 
7 
8 
9 

920 
1 
2 
3 

4 
5 
6 
7 
8 
9 

930 

1 

2 
3 

4 
5 
6 

7 


940 

1 
2 
3 

4 
5 


950 


0123456789 


95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 

5472  5477  5482  5487  5492  5497  5501  5506  5511  551(5 

5521  5525  5530  5535  5540  5545  5550  5554  5559  5504 

5569  5574  5578  5583  5588  5593  5598  5602  5607  5612 

5617  5622  5626  5631  5636  5641  5646  5650  5655  5660 

5665  5670  5674  5679  5684  5689  5694  5698  5703  5708 

5713  5718  5722  5727  5732  5737  5742  5746  5751  575(5 

5761  5766  5770  5775  5780  5785  5789  5794  5799  5804 

5809  5813  5818  5823  5828  5832  5837  5842  5847  5852 

5856  5861  5866  5871  5875  5880  5885  5890  5895  5899 

95904  95909  95914  95918  95923  95928  95933  95938  95942  95947 

5952  5957  5961  5966  5971  5976  5980  5985  5990  5995 

5999  6004  6009  6014  6019  6023  6028  6033  6038  6042 

6047  6052  6057  6061  6066  6071  6076  6080  6085  6090 

6095  6099  6104  6109  6114  6118  6123  6128  6133  6137 

6142  6147  6152  6156  6161  6166  6171  6175  6180  6185 

6190  6194  6199  6204  6209  6213  6218  6223  6227  6232 

6237  6242  6246  6251  6256  6261  6265  6270  6275  6280 

6284  6289  6294  6298  6303  6308  6313  6317  6322  6327 

6332  6336  6341  6346  6350  6355  6360  6365  6369  6374 

<»0379  D(3o84  00388  93303  90398  90402  90407  90412  90417  90421 

6426  6431  6435  6440  6445  6450  6454  6459  6464  6468 

6473  6478  6483  6487  6492  6497  6501  6506  6511  6515 

6520  6525  6530  6534  6539  6544  6548  6553  6558  6562 

6567  6572  6577  6581  6586  6591  6595  6600  6605  6609 

6614  6619  6624  6628  6633  6638  6642  6647  6652  6656 

6661  6666  6670  6675  6680  6685  6689  6694  6699  6703 

6708  6713  6717  6722  6727  6731  6736  6741  6745  6750 

6755  6759  6764  6769  6774  6778  6783  6788  6792  6797 

6802  6806  6811  6816  6820  6825  6830  6834  6839  6844 

96848  96853  96858  96862  96867  96872  96876  96881  96886  96890 

6895  6900  6904  6909  6914  6918  6923  6928  6932  6937 

6942  6946  6951  6956  6960  6965  6970  6974  6979  6984 

6988  6993  6997  7002  7007  7011  7016  7021  7025  7030 

7035  7039  7044  7049  7053  7058  7063  7067  7072  7077 

7081  7086  7090  7095  7100  7104  7109  7114  7118  7123 

7128  7132  7137  7142  7146  7151  7155  7160  7165  7169 

7174  7179  7183  7188  7192  7197  7202  7206  7211  7216 

7220  7225  7230  7234  7239  7243  7248  7253  7257  7262 

7267  7271  7276  7280  7285  7290  7294  7299  7304  7308 

97313  97317  97322  97327  97331  97336  97340  97345  97350  97354 

7359  7364  7368  7373  7377  7382  7387  7391  7396  7400 

7405  7410  7414  7419  7424  7428  7433  7437  7442  7447 

7451  7456  7460  7465  7470  7474  7479  7483  7488  7493 

7497  7502  7506  7511  7516  7520  7525  7529  7534  7539 

7543  7548  7552  7557  7562  7566  7571  7575  7580  7585 

7589  7594  7598  7603  7607  7612  7617  7621  7626  7630 

7635  7640  7644  7649  7653  7658  76(53  7007  7072  7676 

7681  7685  7690  7695  7699  7704  7708  7713  7717  7722 

7727  7731  7736  7740  7745  7749  7754  7759  77(53  7768 

97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 


TABLE  VI.— LOGARITHMS  OF   NUMBERS.  23? 


012345678 


950  97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 

1  7818  7823  7827  7832  7836  7841  7845  7850  7855  7859 

2  7864  7868  7873  7877  7882  7886  7891  7896  7900  7905 

3  7909  7914  7918  7923  7928  7932  7937  7941  7946  7950 

4  7955  7959  7964  7968  7973  7978  7982  7987  7991  7996 

5  8000  8005  8009  8014  8019  8023  8028  8032  8037  8041 

6  8046  8050  8055  8059  8064  8068  8073  8078  8082  8087 

7  8091  8096  8100  8105  8109  8114  8118  8123  8127  8132 

8  8137  8141  8146  8150  8155  8159  8164  8168  8173  8177 

9  8182  8186  8191  8195  8200  8204  8209  8214  8218  8223 

960  98227  98232  98236  98241  98245  98250  98254  98259  98263  98268 

1  8272  8277  8281  8286  8290  8295  8299  8304  8308  8313 

2  8318  8322  8327  8331  8336  8340  8345  8349  8354  8358 

3  8363  8367  8372  8376  8381  8385  8390  8394  8399  8403 

4  8408  8412  8417  8421  8426  8430  8435  8439  8444  8448 

5  8453  8457  8462  8466  8471  8475  8480  8484  8489  8493 

6  8498  8502  8507  8511  8516  8520  8525  8529  8534  8538 

7  8543  8547  8552  8556  8561  8565  8570  8574  8579  8583 

8  8588  8592  8597  8601  8605  8610  8614  8619  8623  8628 

9  8632  8637  8641  8646  8650  8655  8659  8664  8668  8673 

970  98677  98682  98686  98691  98695  98700  98704  98709  98713  98717 

1  8722  8726  8731  8735  8740  8744  8749  8753  8758  8762 

2  8767  8771  8776  8780  8784  8789  8793  8798  8802  8807 

3  8811  8816  8820  8825  8829  8834  8838  8843  8847  8851 

4  8856  8860  8865  8869  8874  8878  8883  8887  8892  8896 

5  8900  8905  8909  8914  8918  8923  8927  8932  8936  8941 

6  8945  8949  8954  8958  8963  8967  8972  8976  8981  8985 

7  8989  8994  8998  9003  9007  9012  9016  9021  9025  9029 

8  9034  9038  9043  9047  9052  9056  9061  9065  9069  9074 

9  9078  9083  9087  9092  9096  9100  9"105  9109  9114  9118 

980  99123  99127  99131  99136  99140  99145  99149  99154  99158  99162 

1  9167  9171  9176  9180  9185  9189  9193  9198  9202  9207 

2  9211  9216  9220  9224  9229  9233  9238  9242  9247  9251 

3  9255  9260  9264  9269  9273  9277  9282  9286  9291  9295 

4  9300  9304  9308  9313  9317  9322  9326  9330  9335  9339 

5  9344  9348  9352  9357  9361  9366  9370  9374  9379  9383 

6  9388  9392  9396  9401  9405  9410  9414  9419  9423  9427 

7  9432  9436  9441  9445  9449  9454  9458  9463  9467  9471 

8  9476  9480  9484  9489  9493  9498  9502  9506  9511  9515 

9  9520  9524  9528  9533  9537  9542  9546  9550  9555  9559 

990  99564  99568  99572  99577  99581  99585  99590  99594  99599  99603 

1  9607  9612  9616  9621  9625  9629  9634  9638  9642  9647 

2  9651  9656  9660  9664  9669  9673  9677  9682  9686  9691 

3  9695  9699  9704  9708  9712  9717  9721  9726  9730  9734 

4  9739  9743  9747  9752  9756  9760  9765  9769  9774  9778 

5  9782  9787  9791  9795  9800  9804  9808  9813  9817  9822 

6  9826  9830  9835  9839  9843  9848  9852  9856  9861  9865 

7  9870  9874  9878  9883  9887  9891  9896  9900  9904  9909 

8  9913  9917  9922  9926  9930  9935  9939  9944  9948  9952 

9  9957  9961  9965  9970  9974  9978  9983  9987  9991  9996 

1000  00000  00004  00009  00013  00017  00022  00026  00030  00035  00039 


238  TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES. 


1° 


Sine       Cosine         Sine         Cosine 


Sine       Cosine 


37 


6.46373 
76476 
94085 

7.06579 
16270 
24188 


36682 
41797 

7.46373 
50512 
54291 
57767 
60985 
63982 
66784 
69417 
71900 
74248 

7.76475 
78594 
80615 
82545 


86166 
87870 
89509 
91088 
92612 

7.94084 
95508 
96887 
98223 
99520 

8.00779 
02002 
03192 
04350 
05478 

8.06578 
07650 


09718 
10717 
11693 
12647 
13581 
14495 
15391 


17128 
17971 
18798 
19610 
20407 
21189 
21958 
22713 
23456 
24186 


10.00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 

10.00000 
00000 
00000 
00000 
00000 
00000 
00000 


9.99999 


99999 
99999 
99999 
99999 


99998 


99997 
99997 

9.99997 
99997 
99997 
99997 


99996 
99996 
99996 
99996 
99996 

.99995 


99995 
99995 
99995 


'.I! »!»'.» I 


99994 
99994 


8.24186 
24903 
25609 
26304 
26988 
27661 
28324 
28977 
29621 
30255 

8.30879 
31495 
32103 
32702 
33292 
33875 
34450 
35018 
35578 
36131 

8.36678 
37217 
37750 
38276 
38796 
39310 
39818 
40320 
40816 
41307 

8.41792 
42272 
42746 
43216 


44139 
44594 
45044 
45489 
45930 


46799 
47226 
47650 


48485 
48896 
49304 
49708 
50108 

8.50504 
50897 
51287 
51673 
52055 
52434 
52810 
53183 
53552 
53919 
54282 


99993 
99992 
99992 
99992 
99992 
99992 


99990 
99990 


5I99S9 


99989 

9.99988 
99988 


99987 
99987 
99986 


99985 


9. 


9998E 


99984 
99984 
99983 


99982 
99982 


99981 
99981 


99980 
99980 
99979 
99979 
99979 
99978 

.99978 
99977 
99977 
99977 
99976 
99976 
99975 
99975 
99974 
99974 
99974 


54642 
54999 
55354 
55705 
56054 
56400 
56743 
57084 
57421 

9.99991   8.57757   9.99969 


58419 
58747 
59072 
59395 
59715 


60349 
60662 


8.60973 

61282 


61894 
62196 
62497 
62795 
63091 
633&5 
63678 

8.63968 
64256 
64543 
64827 
65110 
65391 
65670 
65947 


66497 

8.66769 
67039 
67308 
67575 
67841 
68104 
68367 


5.69400 
69654 
69907 
70159 
70409 
70658 
70905 
71151 
71395 
71638 
71880 


Cosine       Sine 
89° 


Cosine 


Sine 


Cosine 


9.99974 
99973 
99973 
99972 
99972 
69971 
99971 
99970 
99970 


99968 
99967 
99967 
99967 
99966 


99964 

9.99964 
99963 
99963 


.99959 
99958 
99958 
99957 


99956 
99955 
99955 
99954 
99954 


99952 
99952 
99951 


99949 
99948 


9.99947 
99946 
99946 
99945 


99944 


Sine 


88° 


TABLE  VII.— LOGARITHMIC  SINES  AND  COSINES.   239 


o 
i 

2 

3 
4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 


37 


Sine       Cosine         Sine         Cosine 


Sine 


Cosine 


8.71880 
72120 
72359 
72597 
72834 
73069 
73303 
73535 
73767 
73997 

8.74226 
74454 
74680 
74906 
75130 
75353 
75575 
75795 
76015 
76234 

8.76451 
76667 
76883 
77097 
77310 
77522 
77733 
77943 
78152 
78360 

8.78568 
78774 
78979 
79183 
79386 
79588 
79789 
79990 
80189 


8.80585 
80782 


81173 

81367 
81560 
81752 
81944 
82134 
82324 

8.82513 
82701 
82888 
83075 
83261 


83630 
83813 


84177 

84358 


9.99940 
99940 
99939 


99937 


99934 


99932 
99932 


99929 
99929 


99927 


99925 
99924 
99923 
99923 
99922 
99921 


99920 

9.99919 
99918 
99917 
99917 
99916 
99915 
99914 
99913 
99913 
99912 

9.99911 


99907 
99906 


99904 


9.99903 
99902 
99901 
99900 


99897 


8.84358 
84539 
84718 
84897 
85075 
85252 
85429 
85605 
85780 
85955 

J.  86128 
86301 
86474 


86816 
86987 
87156 
87325 
87494 
87661 

.87829 
87995 
88161 


88490 
88654 
88817 


89142 


8.89464 
89625 
89784 
89943 
90102 
90260 
90417 
90574 
90730 
90885 

8.91040 
91195 
91349 
91502 
91655 
91807 
91959 
92110 
92261 
92411 

8.92561 
92710 


93007 
93154 
93301 
93448 


93740 


9.99894 


99884 
99883 
99882 


99863 
99862 
99861 


99857 
.99856 


99854 
99853 
99852 
99851 
99850 
99848 


9.99845 
99844 
99843 


99841 
99840 


99838 
99837 
99836 


8.94030 
94174 
94317 
94461 
94603 
94746 
94887 
95029 
95170 
95310 

8.95450 
95589 
95728 
95867 
96005 
96143 
96280 
96417 
96553 


97095 
97229 
97363 


97629 
97762 


.99866   8.98157 
99865 


98549 
98679 

98808 


99194 


8.99450 
99577 
99704 
99830 
99956 

9.00082 
00207 
00332 
00456 
00581 

9.00704 
00828 
00951 
01074 
01196 
01318 
01410 
01561 
01682 
01803 
01923 


Cosine       Sine 
86° 


Cosine 


Sine 


Cosine 


9.99834 


99832 
99831 


99828 


99825 
99824 


99822 
99821 

99820 


99815 
99814 


99879 
99879 


99877 

9.99876  8.96825  9.99812 
99875 
99874 
99873 
99872 
99871 


99802 
99801 

9.99800 
99798 
99797 
99796 
99795 
99793 
99792 
99791 
99790 


.99787 
99786 
99785 
99783 
99782 
99781 


99778 
99777 
99776 

9.99775 
99773 
99772 
99771 
99769 
99768 
99767 
99765 
99764 


99761 


Sine 


85° 


84C 


240  TABLE  VII.— LOGARITHMIC  SINES  AND  COSINES. 


f 

6° 

7° 

8° 

Sine   Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.01923   9 

.99761 

9.08589 

9.99675 

9.14356 

9.99575 

60 

1 

02043 

99760 

08692 

99674 

14445 

99574 

59 

2 

02163 

99759 

08795 

99672 

14535 

99572 

58 

3 

02283 

99757 

08897 

99670 

146V4 

99570 

57 

4 

02402 

99756 

08999 

99669 

14714 

99568 

56 

5 

02520 

99755 

09101 

99667 

14803 

99566 

55 

6 

02639 

99753 

09202 

99666 

14891 

99565 

54 

7 

02757 

99752 

09304 

99664 

14980 

99563 

53 

8 

02874 

99751 

09405 

99663 

15069 

99561 

52 

9 

02992 

99749 

09506 

99661 

15157 

99559 

51 

10 

9.03109  ,9 

.99748 

9.09606 

9.99659 

9.15245 

9.99557 

50 

11 

03226 

99747 

09707 

99658 

15333 

99556 

49 

12 

03342 

99745 

09807 

99656 

15421 

99554 

48 

13 

03458 

99744 

09907 

99655 

15508 

99552 

47 

14 

03574 

99742 

10006 

99653 

15596 

99550 

46 

15 

03690 

99741 

10106 

99651 

15683 

99548 

45 

16 

03805 

99740 

10205 

99650 

15770 

99546 

44 

17 

03920 

99738 

10304 

99648  • 

15857 

99545 

43 

18 

04034 

99737' 

10402 

99647 

15944 

99543 

•  42 

19 

04149 

99736 

10501 

99645 

16030 

99541 

41 

20 

9.04262   9 

.99734 

9.10599 

9.99643 

9.16116 

9.99539 

40 

21 

04376 

99733 

10697 

99642 

16203 

99537 

39 

22 

04490 

99731 

10795 

99640 

16289 

99535 

38 

23 

04603 

99730 

10893 

99638 

16374 

99533 

37 

24 

04715 

99728 

10990 

99637 

16460 

99532 

36 

25 

04828 

99727 

11087 

99635 

16545 

99530 

35 

26 

04940 

99726 

11184 

99633 

16631 

99528 

34 

27 

05052 

99724 

11281 

99632 

16716 

99526 

33 

28 

05164 

99723 

11377 

99630 

16801 

99524 

32 

29 

05275 

99721 

11474 

99629 

16886 

99522 

31 

30 

9.05386   9 

.99720 

"  9.11570 

9.99627 

9.16970 

9.99520 

30 

31 

05497 

99718 

11666 

99625 

17055 

99518 

29 

32 

05607 

99717 

11761 

99624 

17139 

99517 

28 

33 

05717 

99716 

11857 

99622 

17223 

99515 

27 

34 

05827 

99714 

11952 

99620 

17307 

99513 

26 

35 

05937 

99713 

12047 

99618 

17391 

99511 

25 

36 

06046 

99711 

12142 

99617 

17474 

99509 

24 

37 

06155 

99710 

12236 

99615 

17558 

99507 

23 

38 

06264 

99708 

12331 

99613 

17641 

99505 

22 

39 

06372 

99707 

12425 

99612 

17724 

99503 

21 

40 

9.06481   9 

.99705 

9.12519 

9.99610 

9.17807 

9.99501 

20 

41 

06589 

99704 

12612 

99608 

17890 

99499 

19 

42 

06696 

99702 

12706 

99607 

17973 

99497 

18 

43 

06804 

99701 

12799 

99605 

18055 

99495 

17 

44 

06911 

99699 

12892 

99603 

18137 

99494 

16 

45 

07018 

99698 

12985 

99601 

18220 

99492 

15 

46 

07124 

99696 

13078 

99600 

18:302 

99490 

14 

47 

07231 

99695 

13171 

99598 

18383 

99488 

13 

48 

07337 

99693 

13263 

99596 

18465 

99486 

12 

49 

07442 

99692 

13355 

99595 

18547 

99484 

11 

50 

9.07548   9 

.99690 

9.13447 

9.99593 

9.18628 

9.99482 

10 

51 

07653 

99689 

13539 

99591 

18709 

99480 

9 

52 

07758 

99687 

13630 

99589 

18790 

99478 

8 

53 

07863 

99686 

13722 

99588 

18871 

99476 

y 

54 

07968 

99684 

13813 

99586 

18952 

99474 

6 

55 

08072 

99683 

13904 

99584 

19033 

99472 

5 

56 

08176 

99681 

13994 

99582 

19113 

99470 

4 

57 

08280 

99680 

14085 

99581 

19193 

99468 

3 

58 

08383 

99678 

14175 

99579 

19273 

99466 

2 

59 

08486 

99677 

14266 

99577 

19353 

99464 

1 

60 

08589 

99675 

14356 

99575 

19433 

99462 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

83° 

82° 

81° 

' 

TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES.  241 


9 

10° 

11° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.19433 

9.99462 

9.23967 

9.99335 

9.28060 

9.99195 

60 

1 

19513 

99460 

24039 

99333 

28125 

99192 

59 

2 

19592 

99458 

24110 

99331 

28190 

99190 

58 

3 

19672 

99456 

24181 

99328 

28254 

99187 

57 

4 

19751 

99454 

24253 

99326 

28319 

99185 

56 

5 

19830 

99452 

24324 

99324 

28384 

99182 

55 

6 

19909 

99450 

24395 

99322 

28448 

99180 

54 

7 

19988 

99448 

24466 

99319 

28512 

99177 

53 

8 

20067 

99446 

24536 

99317 

28577 

99175 

52 

9 

20145 

99444 

24607 

99315 

28641 

99172 

51 

10 

9.20223 

9.99442 

9.24677 

9.99313 

9.28705 

9.99170 

50 

11 

20302 

99440 

24748 

99310 

28769 

99167 

49 

12 

20380 

99438 

24818 

99308 

28833 

99165 

48 

13 

20458 

99436 

24888 

99306 

28896 

99162 

47 

14 

20535 

99434 

24958 

99304 

28960 

99160 

46 

15 

20613 

99432 

25028 

99301 

29024 

99157 

45 

16 

20691 

99429 

25098 

99299 

29087 

99155 

44 

17 

20768 

99427 

25168 

99297 

29150 

99152 

43 

18 

20845 

99425 

25237 

99294 

29214 

99150 

42 

19 

20922 

99423 

25307 

99292 

29277 

99147 

41 

20 

9.20999 

9.99421 

9.25376 

9.99290 

9.29340 

9.99145 

40 

21 

21076 

99419 

25445 

99288 

29403 

99142 

39 

22 

21153 

99417 

25514 

99285 

29466 

99140 

38 

28     21329 

99415 

25583 

99283 

29529 

99137 

37 

24     21306 

99413 

25652 

99281 

29591 

99135 

36 

25 

21382 

99411 

25721 

99278 

29654 

99132 

35 

26 

21458 

99409 

25790 

99276 

29716 

99130 

34 

27 

21534 

99407 

25858 

99274 

29779 

99127 

33 

28 

21610 

99404 

25927 

99271 

29841 

99124 

32 

29 

21685 

99402 

25995 

99269 

29903 

99122 

M 

30 

9.21761 

9.99400 

9.26063 

9.99267 

9.29966 

9.99119 

30 

31 

21836 

99398 

26131 

99264 

30028 

99117 

29 

32 

21912 

99396 

26199 

99262 

30090 

99114 

28 

33 

21987 

99394 

26267 

99260 

30151 

99112 

27 

34 

22062 

99392 

26335 

99257 

30213 

99109 

26 

35 

22137 

99390 

26403 

99255 

30275 

99106 

25 

36 

22211 

99388 

26470 

99252 

30336 

99104 

24 

37 

22286 

99:585 

26538 

99250 

30398 

99001 

23 

38 

22361 

99383 

26605 

99248 

30459 

99099 

22 

39 

22435 

99381 

26672 

99245 

30521 

99096 

21 

40 

9.22509 

9.99379 

9.26739 

9.99243 

9.30582 

9.99093 

20 

41 

22583 

99377 

26806 

99241 

30043 

99091 

19 

42 

22657 

99375 

26873 

99-,'38 

30704 

99088    18 

43 

22731 

99372 

26940 

99236 

30765 

9908ti 

17 

44 

22805 

99370 

27007 

99233 

30826 

99083 

16 

45 

22878 

99368 

27073 

99281 

30887 

99080 

15  1 

46 

22952 

99366 

27140 

99229 

30947 

99078 

14 

47 

23025 

99364 

27206 

99226 

31008 

99075 

13 

48 

23098 

99362 

27273 

99224 

31068 

99072 

12 

49 

23171 

99359 

27339 

99221 

31129 

99070 

11 

50 

9.23244 

9.99357 

9.27405 

9.99219 

9.31189 

9.99067 

10 

51 

23317 

99355 

27471 

99217 

31250 

99064 

9 

52 

23390 

99353 

27537 

99214 

31310 

99062 

8 

53 

23462 

99351 

27602 

99212 

31370 

99059 

54 

23535 

99348 

27668 

99209 

31430 

99056 

6 

55 

23607 

99346 

27734 

99207 

31490 

99054 

5 

56 

23679 

99344 

27799 

99204 

81549 

99051 

4 

57 

23752 

99342 

27864 

99202 

31609 

99048 

3 

58 

23823 

99340 

27930 

99200 

31669 

99046 

2 

59 

23895 

99337 

27995 

99197 

31728 

99043 

60 

23967 

99335 

28060 

99195 

31788 

99040 

0 

f 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

80° 

79° 

7 

8° 

242    TABLE   VII.— LOGARITHMIC    SINES   AND   COSINES. 


> 

12°             13° 

14° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.31788 

9.99040 

9.85209 

9.98872 

9.88968 

9.9S690 

60 

1 

31847 

09038 

85263 

98869 

38418 

98687 

59 

2 

31907 

99035 

85318 

98867 

38469 

98684 

58 

3 

31966 

99032 

35373 

9H864 

38519 

98681 

57 

4 

32025 

99030 

35427 

98861 

38570 

•   98678 

56 

5 

32084 

99027 

35481 

98858 

38620 

98675 

55 

6 

32143 

990-24 

35536 

98855 

38670 

98671 

54 

7 

32202 

99022 

35590 

98852 

38721 

98668 

53 

8 

32261 

99019 

35644 

98849 

88771 

98665 

52 

9 

32319 

91)016 

35698 

98846 

38821 

98662 

51 

10 

9.32378 

9.99013 

9.35752 

9.98843 

9.38871 

9.98659 

50 

11 

32437 

99011 

35806 

98840 

38921 

98656 

49 

12 

32495 

99008 

a58GO 

98837 

38971 

98652 

48 

13 

32553 

99005 

35914 

98834 

39021 

98649 

47 

14 

32612 

99002 

35968 

98831 

39071 

98646 

46 

15 

32670 

99000 

36022 

98828 

39121 

98643 

45 

16 

32728 

98997 

36075 

98825 

39170 

98640 

44 

17 

32786 

98994 

36129 

98822 

39220 

98636 

43 

18 

32844 

98991 

36182 

98819 

39270 

98633 

42 

19 

32902 

98989 

36236 

98816 

39319 

86580 

41 

20 

9.32960 

9.98986 

9.36289 

9.98813 

9.39369 

9.98627 

40 

21 

33018 

98983 

36342 

98810 

39418 

98623 

39 

22 

33075 

98980 

36395 

98807 

39467 

98620 

38 

23 

33133 

98978 

36449 

98804 

39517 

98617 

37 

24 

33190 

98975 

36502 

98801 

39566 

98614 

36 

25 

33248 

98972 

36555 

98798 

39615 

98610 

35 

26 

33305 

98969 

36608 

98795 

39664 

98607 

34 

27 

33362 

98967 

36660 

98792 

39713 

98604 

33 

28 

33420 

98964 

36713 

98789 

39762 

98601 

32 

29 

33477 

98961 

36766 

98786 

39811 

98597 

31 

30 

9.33534 

9.98958 

9.36819 

9.98783 

9.39860 

9.98594 

30 

31 

3359  J 

98955 

36871 

98780 

39909 

98591 

29 

32 

33647 

98953 

36924 

98777 

39958 

98588 

28 

33 

33704 

98950 

36976 

98774 

40006 

98584 

27 

34 

33761 

98947 

37028 

98771 

40055 

98581 

26 

35 

33818 

98944 

37081 

98768 

40103 

98578 

25 

36 

33874 

98941 

37133 

98765 

40152 

98574 

24 

37 

33931 

9893S 

37185 

98762 

40200 

98571 

23 

38 

33987 

98936 

37237 

98759 

40249 

98568 

22 

39 

34043 

98933 

37289 

98756 

40297 

985(55 

21 

40 

9.34100 

9.98930 

9.37341 

9.98753 

9.40346 

9.9S561 

20 

41 

34156 

98927 

37393 

98750 

40394 

98558 

19 

42 

34212 

98924 

37445 

98746 

40442 

98555 

18 

43 

34268 

98921 

37497 

98743 

40490 

98551 

17 

44 

343-24 

98919 

37549 

987'40 

40538 

98548 

16 

45 

34380 

98916 

37600 

98737 

40586 

98545 

15  1 

46 

34436 

98913 

37652 

98734 

40634 

98541 

14 

47 

34491 

98910 

37703 

98731 

40682 

98588 

13 

48 

34547 

98907 

37755 

98728 

40730 

98535 

12 

49 

34602 

93904 

37806 

98725 

40778 

98531 

11 

50 

9.34658 

9.98901 

9.37858 

9.98722 

9.40825 

9.98528 

10 

51 

34713 

98898 

37909 

98719 

40873 

98525 

9 

52 

34769 

98896 

37960 

98715 

40921 

98521 

8 

53 

34824 

98893 

3M01  1 

98712 

40968 

98518 

7 

54 

3487!) 

98890 

38062 

98709 

41016 

98515 

6 

55 

34934 

98887 

381  13 

98706 

41063 

98:.  11 

5 

56 

34989 

98884 

88164 

98703 

41111 

98508 

4 

57 

35044 

«>SHS1 

38215 

98700 

41158 

98505 

3 

58 

35099 

98878 

38266 

98697 

41205 

98501 

2 

59 

35154 

98875 

38317 

98694 

41252 

98498 

1 

60 

35209 

98872 

88868 

98690 

41300 

98494 

0 

f 

Cosine 

Sine 

Cosine 

Sim- 

Cosine 

Sine 

77° 

76° 

76° 

TABLE  VII.  — LOGARITHMIC    SINES  AND  COSINES.    243 


lo°             16°              17° 

t 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.41300 

9.98494 

9.44034 

9.98284 

9.16594 

9.98060    60 

1 

41347 

98491 

44078 

98281 

46635 

98056 

59 

2 

41394 

98488 

44122 

98277 

46676 

98052 

58 

3 

41441 

98484 

44166 

98273 

46717 

98048 

57 

4 

41488 

98481 

44210 

98270 

46758 

98044 

56 

5 

41535 

98477 

44253 

98266 

46800 

98040 

55 

6 

41582 

98474 

44297 

98262 

46841 

98036 

54 

7 

41628 

98471 

44341 

98259 

46882 

98032 

53 

8 

41675 

98467 

44385 

98255 

46923 

98029 

52 

9 

41722 

98464 

44428 

98251 

46964 

98025 

51 

10 

9.41768 

9.98460 

9.44472 

9.98248 

9.47005 

9.98021 

50 

11 

41815 

98457 

44516 

98244 

47045 

98017 

49 

12 

41861 

98453 

44559 

98240 

47086 

98013 

48 

13 

41908 

98450 

44602 

98237 

47127 

98009 

47 

14 

41954 

98447 

44646 

98233 

47168 

98005 

46 

15 

42001 

98443 

44689 

98229 

47209 

98001 

45 

16 

42047 

98440 

44733 

98226 

47249 

97997 

44 

17 

42093 

98136 

44776 

98222 

47290 

97993 

43 

18 

42140 

98433 

44819 

98218 

47330 

97989 

42 

19 

42186 

98429 

44862 

98215 

47371 

97986 

41 

20 

9.42232 

9.98426 

9.44905 

9.98211 

9.47411 

9.97982 

40 

21 

42278 

98422 

44948 

98207 

47452 

97978 

39 

22 

42324 

98419 

44992 

98204 

47492 

97974 

38 

23 

42370 

98415 

45035 

98200 

47533 

97970 

37 

24 

42416 

98412 

45077 

98196 

47573 

97966 

36 

25 

42461 

98409 

45120 

98192 

47613 

97962 

35 

26 

42507 

98405 

45163 

98189 

47654 

97958 

34 

27 

42553 

98402 

45206 

98185 

47694 

97954 

33 

28 

42599 

98398 

45249 

98181 

47734 

97950 

32 

29 

42644 

98395 

45292 

98177 

47774 

97946 

31 

30 

9.42690 

9.98391 

9.45334 

9.98174 

9.47814 

9.97942 

30 

31 

42735 

98388 

45377 

98170 

47854 

97938 

29 

32 

42781 

98384 

45419 

98166 

47894 

97934 

28 

33 

42826 

98381 

45462 

98162 

47934 

97930 

27 

34 

42872 

98377 

45504 

98159 

47974 

97926 

26 

35 

42917 

98373 

45547 

98155 

4F014 

97922 

25 

36 

42962 

98370 

45589 

98151 

48054 

97918 

24 

37 

43008 

98366 

45632 

98147 

48094 

97914 

23 

38 

43053 

98363 

45674 

98144 

48133 

97910 

22 

39 

43098 

98359 

45716 

98140 

48173 

97906 

21 

40 

9.43143 

9.98356 

9.45758 

9.98136 

9.48213 

9.97902 

20 

41 

43188 

98352 

45801 

98132 

48252 

97898 

19 

42 

43233 

98349 

45843 

98129 

48292 

97894 

18 

43 

43278 

98345 

45885 

98125 

48332 

97890 

17 

44 

43323 

98342 

45937 

98121 

48371 

97886 

16 

45 

43367 

98338 

45969 

98117 

48411 

97882 

15 

46 

43412 

98334 

46011 

98113 

48450 

97878 

14 

47 

43457 

98331 

46053 

98110 

48490 

97874 

13 

48 

43502 

98327 

46095 

98106 

48529 

97870 

12 

49 

43546 

98324 

46136 

98102 

48568 

97866 

11 

50 

9.43591 

9.98320 

9.46178 

9.98098 

9.48607 

9.97861 

10 

51 

43635 

98317 

46220 

98094 

48647 

97857 

9 

52 

43680 

98313 

46-J62 

98090 

48686 

97853 

8 

53 

43724 

98309 

46303 

98087 

48725 

97849 

7 

54 

43769 

98306 

46345 

98083 

48764 

97845 

6 

55 

43813 

98302 

46386 

98079 

48803 

97841 

5 

56 

43857 

98299 

46428 

98075 

48842 

97837 

4 

57 

43901 

98295 

46469 

98071 

48881 

97833 

3 

58 

43946 

98291 

46511 

98067 

48920 

97829 

2 

59 

43990 

98288 

46552 

98063 

48959 

97825 

1 

60 

44034 

982S4 

46594 

98060 

48998 

97821     0 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine     , 

740 

78° 

72° 

244  TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES. 


t 

18° 

19° 

2 

0° 

Sine 

Cosine 

Siue 

Cosine 

Sine 

Cosine 

0 

9.48998 

9.97821 

9.51264 

9.97567 

9.53405 

9.97299 

60 

1 

49037 

97817 

51301 

97568 

53440 

97294 

59 

2 

49076 

97812 

51338 

97558 

53475 

972S9 

58 

3 

49115 

97808 

51374 

97554 

53509 

97285 

57 

4 

49153 

97804 

51411 

975.50 

53544 

97280 

56 

5 

49192 

97800 

51447 

97545 

53578 

97276 

55 

6 

49231 

97796 

51484 

97541 

53613 

97271 

54 

7 

49269 

97792 

51520 

97536 

53647 

97360 

53 

8 

49308 

97788 

51557 

97532 

53682 

97262 

52 

9 

49347 

97784 

51593 

97528 

53716 

97257 

51 

10 

9.4M85 

9.97779 

9.51629 

9.97523 

9.53751 

9.97252 

50 

It 

49424 

97775 

51666 

97519 

53785 

97248 

49 

12 

49462 

97771 

51702 

97515 

53819 

97243 

48 

13 

49500 

97767 

51738 

97510 

53854 

97238 

47 

14 

49539 

97763 

51774 

97506 

53888 

97234 

46 

15 

49577 

97759 

51811 

97501 

53922 

97229 

45 

16 

49615 

97754 

51847 

97497 

53957 

97224 

44 

17 

49654 

97750 

51883 

97492 

53991 

97220 

43 

18 

49692 

97746 

51919 

97488 

54025 

97215 

42 

19 

49730 

97742 

51955 

97484 

54059 

97210 

41 

20 

9.49768 

9.97738 

9.51991 

9.97479 

9.51093 

9.97206 

40 

21 

49806 

97734 

52027 

97475 

54127 

97201 

39 

22 

49S44 

97729 

52063 

97470 

54161 

97196 

38 

23 

49882 

97725 

52099 

97466 

54195 

97192 

37 

24 

49920 

97721 

52135 

97461 

5422!) 

97187 

36 

25 

49958 

97717 

52171 

97457 

54263 

97182 

» 

26 

49996 

97713 

52207 

97453 

54297 

97178 

34 

27 

50031 

97708 

52242 

97448 

54331 

97173 

33 

28 

50072 

97704 

52278 

97444 

54365 

97168 

32 

29 

50110 

97700 

52314 

97439 

54399 

97163 

31 

30 

9.50148 

9.97696 

9.52350 

9.97435 

9.54433 

9.97159 

30 

31 

50185 

97691 

52385 

97430 

54466 

97154 

29 

32 

50223 

97687 

52421 

97426 

54500 

97149 

28 

33 

50261 

97683 

52456 

97421 

51534 

97145 

27 

34 

50298 

97679 

52492 

97417 

54567 

97140 

26 

35 

50336 

97674 

52527 

97412 

54601 

97135 

25 

36 

50374 

97670 

52563 

97408 

54635 

97130 

24 

37 

50411 

97666 

52598 

97403 

54668 

97126 

23 

38 

50449 

97662 

52634 

97399 

54702 

97121 

22 

39 

50486 

97657 

52669 

97:594 

54735 

97116 

21 

40 

9.50523 

9.97653 

9.52705 

9.97390 

9.54769 

9.97111 

20 

41 

50561 

97649 

52740 

97385 

54802 

97107 

19 

42 

5059S 

97645 

B27T5 

97381 

54836 

97102 

18 

43 

50635 

97640 

52811 

97376 

54869 

97097 

17 

44 

50(573 

97636 

52846 

97372 

54903 

97092 

16 

45 

50710 

97632 

52881 

97367 

54936 

97087 

15 

46 

50747 

97628 

52916 

97363 

54969 

97083 

14 

47 

50784 

97623 

52951 

97856 

55003 

9707S 

13 

48 

50821 

97619 

52986 

97353 

55036 

97073 

12 

49 

50858 

97615 

53021 

97349 

55069 

97068 

11 

50 

9.50896 

9.97610 

9.53056 

9.97344 

9.55102 

9.97'063 

10 

51 

50933 

97606 

53092 

97340 

55136 

97059 

9 

52 

50970 

97602 

53126 

97335 

55169 

97054 

8 

53 

51007 

97597 

53161 

97331 

55202 

97049 

7 

54 

51043 

97698 

53196 

97326 

55235 

97044 

6 

55 

51080 

97589 

53231 

97322 

55268 

97039 

5 

56 

51117 

97584 

53886 

97817 

55301 

97035 

4 

57 

51154 

97580 

53301 

97312 

55334 

97030 

3 

58 

51191 

97576 

53336 

97308 

55307 

1)7025 

2 

59 

51227 

97571 

53370 

97303 

55400 

97020 

1 

60 

51264 

97567 

53405 

97299 

55433 

97015 

0 

i 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

71° 

70° 

09° 

TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES.   245 


t 

21 

° 

22 

jj 

2 

J° 

f 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.55433 

9.97015 

9.57358 

9.96717 

9.59188 

9.96403 

60 

1 

55466 

97010 

573S9 

96711 

59218 

96397 

59 

2 

55499 

97005 

57420 

96706 

59247 

96392 

58 

3 

55532 

97001 

57451 

96701 

59277 

9C3^7 

57 

4 

55564 

96996 

57482 

96696 

53307 

96381 

56 

5 

55597 

96991 

57514 

96691 

59336 

96376 

55 

6 

56680 

96986 

57545 

96686 

59366 

96370 

54 

7 

86603 

9G981 

57576 

96681 

59396 

96365 

53 

8 

55695 

96976 

57607 

96676 

59425 

96360 

52 

9 

55728 

96971 

57638 

96670 

59455 

96454 

51 

10 

9.55761 

9.96966 

9.57669 

9.96665 

9.59484 

9.96349 

50 

11 

55793 

969(52 

57700 

96C60 

59514 

96343 

49 

12 

55826 

96957 

57731 

96655 

59543 

96338 

48 

13 

55858 

9(5952 

57762 

96650 

59573 

96333 

47 

14 

55S91 

96947 

57793 

96645 

59602 

963-^7 

46 

15 

55923 

96942 

57824 

96640 

59632 

96322 

45 

16 

55956 

96937 

57855 

96634 

59661 

96316 

44 

17 

55988 

96932 

57885 

96629 

59690 

96311 

43 

18 

56021 

96927 

57916 

96624 

59720 

96305 

42 

19 

56053 

96922 

57947 

96619 

59749 

96300 

41 

20 

9.50085 

9.96917 

9.57978 

9.96614 

9.59778 

9.96294 

40 

21 

56118 

96912 

58009 

96608 

59808 

96289 

39 

22 

56150 

96907 

58039 

96603 

59837 

96284 

38 

23 

56182 

96903 

58070 

96598 

59866 

96278 

37 

24 

56215 

96898 

58101 

96593 

59895 

96273 

36 

25 

56-247 

96893 

58131 

96588 

59924 

96267 

35 

26 

56279 

96888 

58162 

96582 

59954 

96262 

34 

27 

56311 

96883 

58192 

96577 

59983 

96256 

33 

28 

56343 

90878 

58223 

96572 

60012 

96251 

32 

29 

56375 

96873 

58253 

9G567 

60041 

96245 

31 

30 

9.56408 

9.96868 

9.58284 

9.96562 

9.60070 

9.96240 

30 

31 

56440 

96863 

58314 

96556 

60099 

96234 

29 

32 

56472 

96858 

58345 

96551 

60128 

96229 

28 

33 

56504 

96853 

58375 

96546 

60157 

962','3 

27 

34 

56536 

96848 

58406 

96541 

60186 

8WI8 

26 

35 

56568 

90843 

58436 

96535 

60215 

96212 

25 

36 

56599 

9(5838 

58467 

96530 

60244 

96207 

24 

91 

56631 

96833 

58497 

96525 

60273 

96201 

23 

38 

56663 

96828 

5852? 

96520 

60302 

96196 

22 

39 

56695 

96823 

58557 

96514 

60331 

96190 

21 

40 

9.5(1727 

9.96818 

9.58588 

9.96509 

9.60359 

9.96185 

20 

41 

56759 

96813 

58618 

96504 

60S88 

96179 

19 

42 

56790 

96808 

58648 

96498 

60417 

96174 

18 

43 

56822 

96803 

58678 

96493 

60446 

96168 

17 

44 

56854 

96798 

58709 

96488 

60474 

96162 

16 

45 

56886 

96793 

58739 

9(5483 

60503 

'.'.6157 

15 

46 

56917 

96788 

58769 

96477' 

60532 

96151 

14 

47 

56949 

96783 

58799 

96472 

60561 

96146 

18 

48 

56980 

96778 

58829 

96467 

60589 

96140 

12 

49 

57012 

9(5772 

588r,9 

96461 

60618 

96135 

11 

50 

9.57044 

9.90767 

9.58889 

9.96456 

9.60646 

9.96129 

10 

51 

57075 

967(52 

58919 

90451 

60675 

96128 

9 

52 

57107 

96757 

58949 

96445 

60704 

96118 

H 

53 

57138 

9(5752 

58979 

96440 

60732 

96112 

7 

54 

57169 

96747 

59(109 

96435 

60761 

96107 

6 

55 

57201 

96742 

59039 

96429 

607*9 

96101 

5 

56 

57232 

96737 

59069 

96424 

60818 

96095 

4 

57 

57264 

96732 

59098 

96419 

60846 

96090 

3 

58 

57295 

96727 

59128 

96413 

60875 

960S4 

2 

r,9 

57326 

96722 

59158 

96408 

60903 

96079 

1 

60 

57358 

96717 

59188 

96403 

60931 

96073  |   0 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

68° 

67° 

66° 

246   TABLE  VII.— LOGARITHMIC   SINES   AND  COSINES. 


, 

24° 

25° 

26° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.60931 

9.96073 

9.62395 

9.96738 

9.64184 

9.95866 

60 

1 

609(50 

96067 

62622 

9572.2 

64210 

95860 

59 

0 

0098S 

96062 

62649 

95716 

64230 

95354 

58 

3 

61016 

96056 

62676 

95710 

64202 

95348 

57 

4 

61045 

96050 

62703 

95704 

64288 

95341 

56 

5 

61073 

96045 

62730 

95698 

64313 

95335 

55 

6 

61101 

96039 

62757 

95692 

64339 

95329 

54 

7 

61129 

96034 

62784 

95686 

04:365 

95323 

53 

8 

61158 

9(3028 

62811 

956SO 

64391 

953  17 

52 

9 

61186 

96022 

62838 

95674 

64417 

95310 

51 

10 

9.61214 

9.96017 

9.62865 

9.95668 

9.64442 

9.95304 

50 

11 

61242 

96011 

628:^2 

95663 

64J68 

95298  i  49 

12 

61270 

9(5005 

62918 

95657 

04494 

9521)2 

48 

13 

61298 

96000 

62945 

95651 

64519 

95286 

47 

14 

61326 

95994 

6.  '972 

95645 

64545 

95279 

46 

15 

61354 

95988 

62999 

95639 

64571 

95273 

45 

16 

61382 

95982 

63026 

95633 

64596 

95267 

44 

17 

61411 

95977 

63052 

95627 

04622 

952G1   ;  43 

18 

Til  438 

95971 

63G79 

95621 

64C47 

95254   !  42 

19 

61466 

95965 

63106 

95615 

64673 

95248 

41 

20 

9.61494 

9.95960 

9.63133 

9.95609 

9.64698 

9.95242 

40 

21 

6152.' 

95954 

63159 

95603 

64724 

95336    39 

22 

61550 

95918 

63186 

95597 

64749 

95229    38 

23 

6157S 

95942 

63213 

95591 

G4775 

95223    37 

24 

61606 

95987 

63239 

95585 

64800 

95217    36 

25 

61631 

95931 

63266 

95579 

64826 

95211    35 

26 

61662 

95925 

63292 

95578 

04851 

95204    34 

27 

61IW9 

95920 

633  19  ' 

95567 

C4877 

95198    33 

28 

61717 

95914 

63345 

95561 

64902 

95192 

32 

29 

61745 

95908 

63372 

95555 

64927 

95185 

31 

30 

9.61773 

9.95902 

9.03398 

9.95549 

9.04953 

9.95179 

30 

31 

61800 

95897 

63425 

95543 

64978 

95173    29 

W 

61808 

95891 

63451 

95537 

65003 

95167    28 

33 

61856 

958S5 

63478 

95531 

65029 

95160    27 

34 

61883 

95879 

C3504 

95525 

65054 

95154    26 

35 

61911 

95873 

63531 

95519 

65079 

95148    25 

36 

6193;) 

95868 

63557 

95513 

65104 

95141    24 

37 

61960 

95862 

63588 

95507 

05130 

95135    23 

38 

61994 

95856 

63610 

95500 

65155 

95129    22 

39 

62021 

95850 

63636 

95494 

05  ISO 

95122    21 

40 

9.62049 

9.95844 

9.63662 

9.S3488 

9.65205 

9.95116 

20 

41 

6207'6 

95839 

63689 

95482 

65230 

95110  ,  19 

42 

62104 

95833 

63715 

95476 

65255 

95103    18 

43 

62131 

95827 

63741 

95470 

65281 

95097    17 

14 

62  ing 

95821 

63767 

95464 

65306 

95090   10 

45 

62186 

95815 

63794 

95458 

65331 

95084    15 

46 

62214 

958  10 

63820 

!).->  !52 

05356 

95078    14 

47 

62241 

95804 

63846 

95446 

65381 

95071     13 

48 

622(58 

9571)8 

68372 

95440 

65406 

95065 

12 

49 

62-296 

95792 

63898 

95434 

05431 

95059 

11 

50 

9.62323 

9.95786 

9.63924 

9.95*37 

9.05450 

9.95052 

10 

51 

62350 

95780 

-  63950 

95121 

65481 

95046     9 

52 

62377 

95775 

6397(5 

95415 

65506 

95039  i   8 

r,3 

62405 

95769 

64002 

95409 

65531 

95033     7 

54 

6243-2 

957G3 

04028 

95403 

(55505 

95027     6 

55 

62459 

95757 

64054 

95397 

655HO 

95020     5 

56 

624H6 

95761 

64080 

95391 

05(505 

95014  !   4 

57 

62513 

95745 

64106 

95884 

05630 

95007     3 

58 

62541 

95739 

84182 

93S78 

65655 

!)50(U     2 

r>9 

62568 

93788 

64158 

95872 

65080 

94995     1 

60 

62595 

95728 

64184 

95366 

65705 

94988     0 

f 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine     , 

cr>° 

54* 

68° 

TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES.   247 


27° 

28° 

29° 

> 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.65705 

9.94988 

9.67161 

9.94593 

9.68557 

9.94182 

60 

1 

65729 

94982 

07186 

94587 

68580 

94175 

59 

2 

65754 

94975 

67208 

94580 

68603 

94108 

58 

3 

65779 

94969 

67232 

94573 

68625 

94361 

57 

4 

65804 

94962 

67256 

94567 

68648 

94154 

56 

5 

65828 

94956 

67280 

94560 

68671 

94147 

55 

6 

65853 

94949 

67303 

94553 

68694 

94140 

54 

~ 

65878 

94943 

67327 

94546 

68716 

94133 

53 

8 

65902 

94936 

67350 

94540 

68739 

94126 

52 

9 

65927 

94930 

67374 

94533 

68762 

94119 

51 

10 

9.65952 

9.94923 

9.67398 

9.94526 

9.68784 

9.94112 

50 

11 

65976 

94917 

67421 

94519 

68807 

94105 

44 

12 

66001 

94911 

67445 

94513 

68829 

94098 

48 

13 

06025 

94904 

67468 

94506 

68852 

94090 

47 

14 

66050 

94898 

67492 

94499 

6H*7'5 

94C83 

46 

15 

66075 

94891 

67515 

94492 

68897 

94076 

45 

16 

66099 

94885 

67539 

94485 

68920 

94069 

44 

17 

66124 

94878 

67562 

94479 

68942 

94062 

43 

18 

06148 

94871 

67580 

94472 

68965 

94055 

42 

19 

GG173 

94865 

67609 

94465 

68987 

94048 

41 

20 

9.66197 

9.94858 

9.67633 

9.94458 

9.69010 

9.94041 

40 

21 

66221 

94852 

67656 

94451 

69032 

94034 

39 

22 

66246 

94845 

67680 

94445 

69055 

94027 

38 

23 

66270 

94839 

67708 

94438 

69077 

94020 

37 

24 

66295 

94832 

67726 

94431 

69100 

94012 

36 

25 

66319 

94826 

67750 

94424 

69122 

94005 

35 

26 

66343 

94819 

67773 

94417 

69144 

93998 

34 

27 

66368 

94813 

67796 

94410 

69167 

93991 

33 

28 

66392 

94806 

67820 

94404 

69189 

93984 

32 

29 

66416 

94799 

07843 

94397 

69212 

93977 

31 

30 

9.66441 

9.94793 

9.67866 

9.94390 

9.69234 

9.93970 

30 

31 

66465 

94786 

67890 

94383 

69256 

93963 

29 

32 

66489 

94780 

67913 

94376 

69279 

93955 

28 

33 

66513 

94773 

67936 

94369 

69301 

93948 

27 

34 

66537 

94767 

67959 

94362 

69323 

93941 

26 

35 

06562 

94760 

67982 

94355 

69345 

93934 

25 

36 

665H6 

94753 

68006 

94349 

69368 

93927 

24 

37 

60610 

94747 

08029 

94342 

69390 

93920 

23 

38 

60034 

94740 

68052 

94335 

69412 

93912 

22 

39 

66658 

94734 

68075 

94328 

69434 

93905 

21 

40 

9.66682 

9.94727 

9.68098 

9.94321 

9.69456 

9.93898 

20 

41 

66706 

94720 

68121 

94314 

69479 

93891 

19 

42 

06731 

94714 

68144 

94307 

69501 

93884 

18 

43 

60755 

94707 

68167 

94300 

69523 

93876 

17 

44 

66779 

94700 

68190 

1)4293 

69545 

93869 

16 

45 

66803 

94694 

68213 

94286 

69567 

93862 

15 

46 

66827 

94687 

68237 

94279 

69589 

93855 

14 

47 

66851 

94680 

68260 

94273 

69611 

93847 

18 

48 

66875 

94674 

68888 

94266 

69633 

93840 

12 

49 

66899 

94667 

68305 

94->59 

69655 

93833 

11 

50 

9.66922 

9.94660 

9.68328 

9.94252 

9.69677 

9.93826 

10 

51 

06946 

94654 

68851 

94245 

69699 

93819 

9 

52 

66970 

94647 

68874 

94238 

69721 

93811 

8 

53 

66994 

94C40 

68397 

94231 

69743 

93804 

•j' 

54 

67018 

94034 

68420 

94224 

69765 

93797 

6 

55 

67042 

94627 

68443 

91217 

69787 

937'89 

5 

56 

67066 

94020 

68466 

94210 

69809 

93782 

4 

57 

67090 

94614 

68489 

94203 

(9831 

93775 

3 

58 

67113 

94607 

68512 

94196 

69853 

93768 

2 

59 

67137 

94000 

68584 

9J1S9 

09875 

93760 

1 

00 

67101 

94593 

68557 

94182 

69897 

93753 

0 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine     , 

«2° 

61° 

60° 

248  TABLE  VII.—  LOGARITHMIC   SINES  AND  COSINES. 


30° 

31° 

32° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.69897 

9.93753 

9.71184 

9.93307 

9.72421 

9.92842 

60 

1 

69919 

93746 

71205 

93099 

72441 

92834 

59 

2 

69941 

93738 

71226 

93291 

72461 

92826 

58 

3 

69963 

93731 

71247 

93284 

72482 

92818 

57 

4 

69984 

93724 

71268 

93076 

72502 

92810 

56 

5 

70006 

93717 

71289 

93069 

70522 

92803 

55 

6 

70028 

93709 

71310 

93261 

70540 

92795 

54 

70050 

93702 

71331 

93053 

70560 

90787 

53 

8 

70072 

U8S95 

71350 

93246 

72582 

90779 

52 

9 

70093 

93687 

71373 

93238 

70602 

92771 

51 

10 

9.70115 

9.93680 

9.71393 

9.93230 

9.72622 

9.90763 

50 

11 

70137 

93673 

71414 

93&S 

70643 

92755 

49 

12 

70159 

93665 

71435 

93215 

72663 

90747 

48 

13 

70180 

93658 

71456 

93007 

72683 

92739 

47 

14 

70202 

93650 

71477 

93000 

72703 

92731 

46 

15 

70224 

93643 

71498 

93192 

72723 

92723 

45 

16 

70245 

93636 

71519 

93184 

72743 

90715 

44 

17 

70067 

93628 

71539 

93177 

72763 

92707 

43 

18 

70288 

93(501 

71560 

93169 

72783 

90699 

42 

19 

70310 

93614 

71581 

93161 

72803 

92691 

41 

20 

9.70332 

9.93606 

9.71602 

9.93154 

9.72823 

9.92683 

40 

21 

7'0353 

93599 

71622 

93146 

70843 

98675 

39 

22 

70375 

93591 

71643 

93138 

72863 

92667 

38 

70396 

93584 

71664 

93131 

72883 

90659 

37 

24 

70418 

93577 

71685 

93123 

72900 

90651 

36 

25 

70439 

93569 

71705 

93115 

72922 

92643 

35 

26 

704(51 

93562 

71726 

93108 

72942 

92635 

34 

27 

70482 

93554 

71T47 

93100 

72988 

92(527 

33 

28 

70504 

93547 

71767 

93092 

70982 

90619 

32 

29 

70525 

93539 

71788 

93084 

73002 

92611 

31 

30 

9.70547 

9.93532 

9.71809 

9.93077 

9.73022 

9.92603 

30 

31 

705)58 

93525 

71829 

93069 

73041 

92595 

29 

32 

70590 

93517 

71850 

93061 

73061 

925S7 

28 

33 

70611 

93510 

71870 

93053 

73081 

92579 

27 

34 

70633 

93502 

71891 

93046 

73101 

92571 

26 

35 

70654 

93495 

71911 

93038 

73121 

90563 

25 

36 

70675 

93487 

71932 

93030 

73140 

92555 

24 

37 

70697 

93  1*0 

71952 

93002 

73160 

90546 

23 

38 

70718 

93172 

71973 

93014 

73180 

90538 

OO 

39 

70739 

934(55 

71994 

93007 

73000 

92530 

21 

40 

9.70761 

9.93457 

9.72014 

9.92999 

9.73219 

9.92522 

20 

41 

70780 

93450 

72034 

90991 

73239 

92514 

19 

42 

70803 

93142 

72055 

92983 

73059 

92506 

18 

43 

70824 

934-55 

72075 

92976 

73078 

90498 

17 

44 

70846 

93427 

72096 

92968 

73298 

90490 

16 

45 

70867 

93420 

72116 

90960 

73318 

92482 

15 

46 

78888 

93412 

70137 

92952 

73337 

92473 

14 

47 

70909 

93405 

72157 

92944 

78857 

92465 

13 

48 

70931 

93397 

72177 

92936 

73377 

92457 

12 

49 

70952 

93390 

72198 

9  01)09 

73396 

92449 

11 

50 

9.70973 

9.93382 

9.72218 

9.92921 

9.73416 

9.92441 

10 

51 

70994 

93375 

70238 

92913 

73435 

92433 

9 

52 

71015 

93387 

72259 

90905 

73455 

90425 

8 

53 

71036 

93360 

72**79 

92897 

73474 

92416 

54 

71058 

93352 

72099 

90SK9 

73494 

92408 

6 

55 

71079 

93344 

72320 

92881 

73513 

90400 

5 

56 

71100 

93337 

72340 

92874 

73533 

90392 

4 

57 

71121 

93329 

72360 

92866 

73550 

90384 

3 

58 

71142 

93322 

72381 

92858 

73572 

903'  6     2 

59 

71163 

93314 

72401 

92850 

73591 

90367 

1 

60 

71184 

93307 

72421 

92842 

73611 

90359 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

5»° 

58° 

57° 

TABLE  TIT.— LOGARITHMIC   SINES  AND  COSINES. 


1  , 

33°             84°             3o° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.73611 

9.92359 

9.74756 

9.9!8r>7 

9.75859 

9.91336 

60 

1 

73G30 

92351 

74775 

91849 

75877 

91328 

59 

2 

73C50 

92343 

74794 

91840 

75895 

91319 

58 

3 

73609 

92335 

74812 

91832 

75913 

91310 

57 

4 

73689 

92326 

74831 

91823 

75931 

91301 

56 

5 

73708 

92318 

74850 

91815 

75949 

91292 

55 

6 

78727 

92310 

74868 

91806 

75967 

91283 

54 

7 

7*747 

92302 

74887 

91798 

75985 

91274 

53 

8 

73766 

92293 

74906 

91789 

76003 

91266 

52 

9 

73785 

92285 

74S24 

91781 

76021 

91257 

51 

10 

9.73805 

9.92277 

9.74943 

9.91772 

9.76039 

9.91248 

50 

II 

73824 

92269 

74961 

91763 

7(5057 

91239 

49 

ts 

73843 

922CO 

74980 

91755 

76075 

91230 

48 

13 

73863 

92252 

74999 

91746 

76093 

91221 

47 

14 

73882 

92244 

75017 

91738 

76111 

91212 

46 

15 

73901 

92235 

7f>036 

91729 

76129 

91203 

45 

16 

73921 

92227 

75054 

91720 

76146 

91194 

44 

17 

73940 

92219 

75073 

91712 

76164 

91185 

43 

18 

73959 

92211 

75091 

91703 

76182 

91176 

42 

19 

73978 

92202 

75110 

91695 

76200 

91167 

41 

20 

9.73997 

9.92194 

9.75128 

9.  91  ('86 

9.76218 

9.91158 

40 

21 

74017 

92186 

75147 

91677 

76236 

91149 

39 

22 

74036 

92177 

75165 

91669 

76253 

91141 

38 

23 

74055 

92169 

75184 

91660 

76271 

91132 

37 

24 

74074 

92161 

75->02 

91651 

76289 

91123 

36 

25 

74093 

92152 

75221 

91643 

76307 

91114 

35 

26 

74113 

92144 

75239 

916.^4 

76324 

91  105 

34 

27 

74132 

92136 

75258 

91625 

76342 

91096 

33 

28 

74151 

92127 

75276 

91617 

76360 

91087 

32 

29 

74170 

92119 

75294 

91608 

76378 

91078 

31 

30 

9.74189 

9.92111 

9.75313 

9.91599 

9.76395 

9.91069 

30 

3! 

74208 

92102 

75331 

91591 

76413 

91060 

29 

32 

74227 

92094 

75350 

91582 

76431 

91051 

28 

33 

74246 

92086 

75368 

91573 

76448 

91042 

27 

34 

74265 

92077 

75386 

91565 

76466 

91033 

26 

35 

74284 

92069 

75405 

91556 

76484 

91023 

25 

36 

74303 

92060 

75423 

91547 

76501 

91014 

24 

37 

74322 

92052 

75441 

91538 

76519 

91005 

23 

38 

74341 

92044 

75459 

91530 

76537 

90996 

22 

39 

74360 

92035 

75478 

91521 

76554 

90987 

21 

40 

9.7437S 

9.92027 

9.75496 

9.91512 

9.76572 

9.90978 

20 

41 

74398 

92018 

75514 

91504 

76590 

90969 

19 

42 

74417 

92010 

75533 

91495 

76607 

90960 

18 

43 

74436 

92002 

75551 

91486 

7«625 

90951 

17 

44 

74455 

91993 

75569 

91477 

76642 

90942 

16 

4.r> 

74474 

91985 

75587 

91469 

76660 

90933 

15 

46 

74493 

91976 

75605 

91460 

76677 

90924 

14 

47 

74512 

91968 

75624 

91451 

76695 

90915 

13 

48 

74531 

91959 

75642 

91442 

76712 

90906 

12 

49 

74549 

91951 

75660 

91433 

76730 

90896 

11 

50 

9.74568 

9.91942 

9.75678 

9.91425 

9.76747 

9.90887 

10 

51 

74587 

91934 

75696 

91416 

76765 

90878 

9 

52 

74606 

91925 

75714 

91407 

76782 

90869 

8 

53 

74625 

91917 

75733 

91398 

76800 

90860 

54 

74644 

91908 

75751 

91389 

76817 

90851 

6 

55 

74662 

91900 

75769 

91381 

76835 

90842 

5 

56 

74681 

91891 

75787 

91372 

76852 

90832 

4 

57 

74700 

91883 

75805 

91363 

76870 

90823 

3 

58 

74719 

91874 

75823 

91354 

76887 

90814 

2 

59 

74737 

91866 

75841 

91345 

76904 

90805 

1 

60 

74756 

91857 

75859 

91336 

76922 

90796 

0 

f 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

66° 

65° 

54° 

250  TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES. 


$ 

36° 

37° 

38° 

Sine 

Cosine 

Sine 

Cosiue 

Sine 

Cosine 

0 

9.76922 

9.90796 

9.77946 

9.90235 

9.78934 

9.89653 

60 

1 

70939 

9078? 

77H63 

90225 

78950 

89643 

59 

2 

76957 

90777 

77980 

90216 

78967 

89633 

58 

3 

76974 

90768 

77997 

90206 

78983 

89624 

57 

4 

76991 

90759 

78013 

90197 

78999 

89614 

56 

5 

77009 

90750 

78030 

90187 

79015 

89604 

55 

6 

77026 

90741 

78047 

90178 

79031 

89594 

54 

7 

77043 

90731 

78063 

90168 

79047 

89584 

53 

8 

77061 

90722 

78080 

90159 

79063 

89574 

52 

9 

77078 

90713 

78097 

90149 

79079 

89564 

51 

10 

9.77095 

9.90704 

9.78113 

9.90139 

9.79095 

9.89554 

50 

11 

77112 

90694 

78130 

90130 

79111 

89544 

49 

12 

77130 

90685 

78147 

90120 

79128 

89534 

48 

13 

77147 

90676 

78163 

90111 

79144 

89524 

47 

14 

77164 

90667 

78180 

90101 

79160 

89514 

46 

15 

77181 

90657 

78197 

90091 

79176 

89504 

45 

16 

77199 

90648 

78213 

90082 

79192 

89495 

44 

17 

77216 

90639 

78230 

90072 

79208 

89485 

43 

18 

77233 

90630 

78246 

90063 

79224 

89475 

42 

19 

77250 

90620 

78263 

90053 

79240 

89465 

41 

20 

9.77268 

9.90611 

9.7S280 

9.90043 

9.79256 

9.89455 

40 

21 

77285 

90602 

78296 

90034 

79272 

89445 

39 

22 

77302 

90592 

78313 

90024 

79288 

89435 

38 

523 

77319 

90583 

78329 

90014 

79304 

89425 

37 

24 

77336 

90574 

78346 

90005 

79319 

89415 

36 

25 

77353 

90565 

78362 

89995 

79335 

89405 

35 

26 

77370 

90555 

78379 

89985 

79351 

89395 

34 

27 

77387 

90546 

78395 

89976 

79367 

89385 

89 

28 

77405 

90537 

78412 

89966 

79383 

89375 

32 

29 

77422 

90527 

78428 

89956 

79399 

89364 

31 

30 

9.77439 

9.90518 

9.78445 

9.89947 

9.79415 

9.89354 

30 

31 

77456 

90509 

78481 

89937 

79431 

89344 

29 

32 

77473 

90499 

78478 

89927 

79447 

89334 

28 

33 

77190 

90490 

78494 

89918 

79463 

89324 

27 

34 

77507 

90480 

78510 

R9908 

79478 

89314 

26 

35 

77524 

90471 

78527 

89898 

79494 

89301 

25 

36 

77541 

90462 

78543 

89888 

79510 

89294 

24 

37 

77558 

90452 

78560 

89879 

79526 

89284 

23 

38 

77575 

90443 

78576 

89869 

79542 

89274 

22 

39 

77592 

90434 

78592 

89859 

79558 

89264 

21 

40 

9.77609 

9.90424 

9.78609 

9.89849 

9.79573 

9.89254 

20 

41 

77626 

90415 

78625 

89840 

79589 

89244 

19 

42 

77643 

90405 

78642 

89830 

79605 

89333 

18 

43 

77660 

90396 

78658 

89820 

78621 

89223 

17 

44 

77677 

90386 

78674 

89810 

79636 

89213 

16 

45 

77694 

90377 

78691 

89801 

79652 

89203 

15 

46 

77711 

90368 

78707 

89791 

79668 

89193 

14 

47 

77728 

90358 

78723 

897'81 

79684 

89183 

13 

48 

77744 

90349 

78739 

89771 

79699 

89173 

12 

49 

77701 

90339 

78756 

89761 

79715 

89162 

11 

50 

9.77778 

9.90330 

0.78772 

9.89752 

9.79731 

9.89152 

10 

51 

77795 

90320 

78788 

89742 

79746 

89142 

9 

52 

77812 

90311 

78805 

89732 

79762 

89132 

8 

53 

778^9 

90301 

78821 

89722 

79778 

89122 

7 

54 

77846 

90292 

78837 

89712 

79798 

89112 

6 

55 

77862 

90282 

78853 

89702 

79809 

89101 

5 

56 

77879 

90273 

78869 

89693 

79825 

89091 

4 

57 

77896 

90263 

78886 

89683 

79840 

89081 

3 

58 

77913 

90254 

78902 

89673 

79856 

89071 

0 

59 

77930 

90244 

78918 

89663 

79872 

89060 

1 

60 

77946 

90235 

78934 

89653 

79887 

ggono 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

58° 

62° 

51° 

TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES.   251 


39° 

40° 

41° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.79887 

9.89050 

9.80807 

9.88425 

9.81694 

9.87778    60 

1 

79903 

89040 

80822 

88415 

81709 

87767    59 

2 

79918 

89030 

80837 

88404 

81723 

87756 

58 

3 

79934 

89020 

80852 

88394 

81738 

87745 

57 

4 

79950 

89009 

80867 

88383 

81752 

87734 

56 

5 

79965 

88999 

80882 

88372 

81767 

87723 

55 

6 

79981 

88989 

80897 

88362 

81781 

87712 

54 

7 

79996 

88978 

80912 

88351 

81796 

877'01 

53 

8 

80012 

88968 

80927 

88340 

81810 

87690 

52 

9 

80027 

88958 

80942 

88330 

81825 

87679    51 

10 

9.80043 

9.88948 

9.80957 

9.88319 

9.81839 

9.87668    50 

11 

80058 

8H937 

80972 

88308 

81854 

87657  '  49 

12 

80074 

88927 

80987 

88298 

81868 

87646    48 

13 

80089 

88917 

81002 

88287 

81882 

87635  i  47 

14 

80105 

88906 

81017 

88276 

81897 

87624 

46 

15 

80120 

88896 

81032 

88266 

81911 

87613 

45 

16 

80136 

88886 

81047 

88255 

81926 

87601 

44 

17 

80151 

88875 

81061 

88244 

81940 

87590 

43 

18 

80166 

88865 

81076 

88234 

81955 

87579 

42 

19 

80182 

88855 

81091 

88223 

81969 

87568 

41 

20 

9.80197 

9.88844 

9.81106 

9.88212 

9.81983 

9.87557 

40 

jjfj 

80213 

88834 

81121 

88201 

81998 

87546 

39 

22 

80228 

88824 

81136 

88191 

82012 

87535 

38 

23 

80244 

88813 

81151 

88180 

82026 

87524 

37 

24 

80259 

8S803 

81166 

88169 

82041 

87513 

36 

25 

80274 

88793 

81180 

88158 

820.-5 

87501 

35 

26 

80290 

88782 

81195 

88148 

82069 

87490 

34 

27 

80305 

88772 

81210 

88137 

82084 

87479 

33 

28 

80320 

88761 

81225 

88*26 

82098 

87468 

32 

29 

80336 

88751 

81240 

88115 

82112 

87457 

31 

30 

9.80351 

9.88741 

9.81254 

9.88105 

9.82126 

9.87446 

30 

31 

80366 

88730 

81209 

88094 

82141 

87434 

29 

32 

80382 

88720 

812H4 

88083 

82155 

87423 

28 

33 

80397 

88709 

81299 

88072 

82169 

87412 

27 

34 

80412 

88699 

81314 

88061 

82184 

87401 

26 

35 

80428 

88688 

81328 

88051 

82198 

87390 

25 

36 

80443 

88678 

81343 

88040 

82212 

87378 

24 

37 

80458 

88668 

81358 

88029 

82226 

87367 

23 

38 

80-173 

88657 

81372 

88018 

82240 

87356 

22 

39 

80489 

88647 

81387 

88007 

82255 

87345 

21 

40 

9.80504 

9.88636 

9.81402 

9.87996 

9.82269 

9.87334 

20 

41 

80519 

88(526 

81417 

87985 

82283 

87322    19 

42 

80534 

88615 

81431 

87975 

82297 

87311 

18 

43 

80550 

88605 

81446 

87964 

82311 

87300 

17 

44 

80565 

88594 

81461 

87953 

82326 

87288    16 

45 

80580 

88584 

8147'5 

87942 

82340 

87277    15 

46 

80595 

8S573 

81490 

87931 

8v354 

87266  I  14 

47 

80010 

88563 

81505 

87920 

82368 

87255    13 

48 

80625 

88552 

81519 

87909 

82382 

87243 

12 

49 

80641 

88542 

81534 

87898 

82396 

87232 

11 

50 

9.80656 

9.88531 

9.81549 

9.87887 

9.82410 

9.87221 

10 

51 

80671 

88521 

8T663 

87877 

82424 

87209 

9 

52 

80686 

88510 

81578 

87866 

82439 

87198 

8 

53 

80701 

88499 

81592 

87855 

824:"  3 

87187 

7 

54 

80716 

88489 

81607 

87844 

82467 

87175 

6 

55 

80731 

88478 

81622 

87833 

82481 

87164 

5 

56 

80746 

88468 

81636 

87822 

82495 

87153 

4 

57 

80762 

88457 

81651 

87811 

82509 

87141 

3 

58 

80777 

88447 

8166S 

87800 

82523 

87130 

2 

59 

80792 

88436 

81680 

877'89 

82537 

87119 

1 

60 

80807 

88425 

81694 

87778 

82551 

87107 

0 

f 

Cosine 

Sine 

CoSfne 

Sine 

Cosine 

•   Sine 

t  • 

60° 

49° 

48° 

TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES. 


/ 

42° 

43° 

44° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.82551 

9.8710T 

9.88378 

9.86413 

9.84177 

9.85G93 

60 

8-25(55 

87096 

83392 

86401 

84190 

85681 

59 

2 

82579 

87035 

83405 

80389 

84203 

85009 

58 

3 

82593 

87073 

83419 

86377 

84216 

85657 

57 

4 

82607 

87062 

83432 

86366 

84229 

85G45 

56 

5 

82621 

87050 

83446 

86354 

84242 

85682 

55 

6 

82635 

87039 

83459 

86342 

84255 

85020 

54 

7 

82649 

87028 

83473 

80330 

84269 

85608 

53 

8 

82663 

87016 

83486 

86318 

84282 

85596 

52 

9 

82677 

87005 

83500 

86306 

84295 

85583 

51 

10 

9.82691 

9.86993 

9.83513 

9.86295 

9.84308 

9.85571 

50 

11 

82705 

86982 

83527 

86283 

84321 

85559 

49 

12 

82719 

86970 

83540 

86271 

84334 

85547 

48 

13 

82733 

86959 

83554 

86259 

84347 

85534 

47 

14 

82747 

86947 

83507 

86247 

84360 

85522 

46 

15 

8«rei 

86936 

83581 

86235 

84373 

85510 

45 

16 

82775 

86924 

83594 

86223 

84385 

85497 

44 

17 

82788 

86913 

83608 

86211 

84398 

85485 

43 

18 

82802 

8C902 

88631 

86200 

84411 

85473 

42 

19 

82816 

86890 

83634 

86188 

84424 

85460 

41 

20 

9.82830 

9.86879 

9.83648 

9.86176 

9.84437 

9.85448 

40 

21 

82844 

86867 

83661 

86164 

84450 

85436 

39 

22 

82858 

86855 

83674 

86152 

84463 

85423 

38 

23 

82872 

8(5844 

83688 

86140 

84476 

85411 

37 

24 

82885 

86832 

83701 

86128 

84489 

85399 

30 

25 

82899 

86821 

83715 

86116 

84502 

85386 

35 

26 

82913 

86809 

88728 

86104 

84515 

85374 

34 

27 

82927 

86798 

83741 

86092 

84528 

85301 

33 

28 

82941 

86786 

83755 

86080 

84540 

85349 

32 

29 

82955 

86775 

83768 

86068 

84553 

85337 

ai 

30 

9.82968 

9.86763 

9.83781 

9.86056 

9.84566 

9.85324 

30 

31 

82982 

86752 

83795 

86044 

84579 

85312 

29 

32 

82996 

86740 

83808 

86032 

84592 

85299 

28 

33 

83010 

86728 

83821 

86020 

84605 

85287 

27 

34 

83023 

86717 

83834 

86008 

84618 

85274 

26 

35 

83037 

86705 

83848 

85996 

84630 

85262 

25 

36 

83051 

86694 

83861 

85984 

84643 

85250 

24 

37 

83065 

86682 

83874 

85972 

84656 

85237 

23 

38 

83078 

86670 

83887 

85900 

84669 

85225 

22 

39 

83092 

86659 

83901 

85948 

84682 

85212 

21 

40 

9.83106 

9.86647 

9.83914 

9.85936 

9.84694 

9.85200 

20 

41 

83120 

86635 

83927 

85924 

84707 

85187 

19 

42 

83133 

86624 

83910 

85912 

84720 

85175 

18 

43 

83147 

86612 

83954 

85900 

84733 

85162 

17 

44 

83161 

86600 

83967 

85888 

84745 

85150 

16 

45 

83174 

86589 

83980 

85876 

84758 

85137 

15  1 

46 

83188 

86577 

83993 

85864 

84771 

85125 

14 

47 

83202 

86565 

84006 

85851 

84784 

85112 

13 

48 

83215 

86554 

84020 

85839 

84796 

85100 

12 

49 

83229 

86542 

84033 

85827 

84809 

85087 

11 

50 

9.83242 

9.86530 

9.84046 

9.85815 

9.  8  4  822 

9.85074 

10 

51 

83256 

86518 

84059 

85S03 

84835 

85062 

9 

52 

83270 

86507 

84072 

85791 

84847 

85049 

8 

53 

83283 

86495 

84085 

85779 

84860 

85037 

t 

54 

83297 

86483 

84098 

85766 

84873 

85024 

6 

55 

88310 

86472 

84112 

85754 

84885 

85012 

5 

56 

83324 

86460 

84  125 

85742 

84898 

84999 

4 

57 

83338 

86448 

84138 

85730 

84911 

84986 

3 

58 

833-.1 

86436 

84151 

85718 

s  »•.)•_>:{ 

84974 

0 

59 

83365 

86425 

841  (',4 

85706 

84936 

84961 

1 

60 

83378 

86413 

84177 

85693 

84949 

84949 

0 

/ 

Gostae 

•  Sine 

Cosine 

Sine 

Cosine 

Sine 

t 

47° 

46° 

46° 

TABLE  VIII.— LOG.  TANGENTS    AND   COTANGENTS.    253 


0° 

1° 

2° 

/ 

Tan 

Cotan 

Tan 

Co  (an 

Tan 

Cotan 

o 

—  CO 

00 

8.24192 

11.75808 

8.  5181  IS 

11.45692 

60 

1 

6.46373 

13.53627 

24910 

75090 

54669 

45331 

59 

76476 

23524 

25616 

74384 

55027 

44973 

58 

3 

94085 

05915 

26312 

73688 

55382 

44618 

57 

4 

7.06579 

12.93421 

26996 

73004 

55734 

44266 

56 

5 

16270 

83730 

27669 

72331 

56083 

43917 

55 

6 

24188 

75812 

28332 

71668 

56429 

43571 

54 

7 

30882 

69118 

28986 

71014 

56773 

43227 

53 

8 

36682 

63318 

29629 

70371 

57114 

42S86 

52 

9 

41797 

5SJ03 

30263 

69737 

57452 

42548 

51 

10 

7.46373 

12.53627 

8.30888 

11.69112 

8.57788 

11.42212 

50 

11 

50512 

49188 

31505 

68495 

58121 

41879  j  49 

12 

54291 

45709 

32112 

67888 

58451 

41549    48 

13 

57767 

42233 

32711 

67289 

58779 

41221    47 

14 

60986 

39014 

33302 

66698 

59105 

40895    46 

15 

63982 

36018 

33886 

66114 

59428 

40572    45 

16 

66785 

33215 

34461 

65539 

59749 

-  40251    44 

17 

69418 

30582 

35029 

64971 

60068 

39932    43 

18 

71900 

28100 

35590 

64410 

60384 

39616    42 

19 

74248 

25752 

36143 

63857 

60698 

39302    41 

20 

7.76476 

12.23524 

8.36689 

11.63311 

8.61009 

11.38991 

40 

21 

78595 

21405 

37229 

62771 

61319 

38681 

39 

22 

80615 

19385 

37762 

62-J38 

61626 

38374  1  38 

23 

82546 

17454 

38289 

61711 

61931 

3H069  !  37 

24 

84394 

15606 

38809 

61191 

62234 

37766  i  36 

25 

86167 

13833 

39323 

60677 

62535 

37465  :  35 

26 

87871 

12129 

39832 

60168 

62834 

37166 

34 

27 

89510 

10490 

40334 

59666 

63131 

30869 

33 

28 

91089 

08911 

40830 

59170 

63426 

36574 

32 

29 

92613 

07387 

41321 

58679 

63718 

36282 

31 

30 

7.94086 

12.05914 

8.41807 

11.58193 

8.64009 

11.35991 

30 

31 

95510 

04490 

42287 

57713 

64298 

35702 

29 

32 

96889 

03111 

42762 

57238 

64585 

35415 

28 

33 

98225 

01775 

43232 

56768 

64870 

35130 

"   M1? 

34 

99522 

00478 

43096 

56304 

65154 

34846 

26 

35 

8.00781 

11.99219 

44156 

55844 

65435 

31565 

25 

36 

02004 

97996 

44611 

55389 

65715 

34285 

24 

37 

03194 

9GS06 

45061 

54939 

65993 

34007 

23 

38 

0-1353 

95647 

45507 

54493 

66269 

33731 

22 

39 

05481 

94519 

45948 

54052 

66543 

33457 

21 

40 

8.06581 

11.93419 

8.46385 

11.53615 

8.66816 

11.  553184 

20 

41 

07653 

92347 

46817 

531  83 

67087 

32913 

19 

42 

08700 

91300 

47215 

52755 

67356 

32644 

18 

43 

097-J2 

90278 

47669 

52331 

67624 

32376 

17 

44 

10720 

89280 

48089 

51911 

67890 

32110 

16 

45 

11696 

88304 

48505 

51495 

68154 

31846 

15 

46 

12651 

87349 

48917 

51C83 

68417 

31583 

14 

47 

13585 

86415 

49325 

50675 

68678 

31322 

13 

48 

14500 

85500 

49729 

50271 

68938 

31062 

12 

49 

15395 

84605 

50130 

49870 

69196 

30804 

11 

50 

8.16*73 

11.83727 

8.50527 

11.49473 

8.69453 

11.30547 

10 

51 

17133 

82867 

50920 

49080 

69708 

30292 

9 

52 

179T6 

82024 

51310 

48690 

69962 

30038 

8 

53 

18804 

81196 

51696 

48304 

70214 

29786 

7 

54 

19616 

803S4 

52079 

47921 

70465 

29535 

6 

55 

20413 

79587 

52459 

47541 

70714 

29286 

5 

56 

21195 

78805 

52835 

47165 

70962 

29038 

4 

57 

21964 

78036 

53208 

46792 

71208 

28792 

3 

58 

22720 

77280 

53578 

46422 

71453 

2H547 

2 

59 

23462 

76538 

53945 

46055 

71697 

28303 

1 

60 

24192 

75808 

54308 

45692 

71940 

28060 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

8!»° 

88° 

87° 

254  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


8° 

4° 

5° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

8.71940 

11.28060 

8.84464 

11.15536 

8.94195 

11.05805    60 

1 

72181 

27819 

84646 

15354 

94340 

05660 

59 

2 

72420 

27580 

84826 

15174 

94485 

05515 

58 

3 

72659 

27341 

85006 

14994 

94C30 

05370 

57 

4 

7289(5 

27104 

85185 

14815 

94773 

05227 

56 

5 

73132 

26868 

85363 

14637 

94917 

05083 

55 

6 

73366 

26634 

85540 

14460 

95060 

04940 

54 

7 

73600 

26400 

85717 

14283 

95202 

04798 

53 

8 

73832 

2616S 

&5893 

14107 

95344 

04656 

52 

9 

74063 

25937 

86069 

13931 

95486 

04514 

51 

10 

8.74292 

11.25708 

8.86243 

11.13757 

8.95627 

11.04373 

50 

11 

74521 

25479 

86417 

13583 

95767 

04233 

49 

IB 

74748 

25252 

86591 

13409 

95908 

04092 

48 

13 

74974 

25026 

86763 

13237 

96047 

03953 

47 

14 

75199 

24801 

86935 

13065 

96187 

03813 

46 

15 

.  75423 

24577 

87106 

12894 

96325 

03075 

45 

16 

75645 

24355 

87277 

12723 

96464 

03536 

44 

17 

75867 

24133 

87447 

12553 

96602 

03398 

43 

18 

76087 

23913 

87616 

12384 

96739 

03261 

42 

19 

76306 

23694 

87785 

12215 

96877 

02123 

41 

20 

8.76525 

11.23475 

8.87953 

11.12047 

8.97013 

11.02987 

40 

21 

76742 

23258 

88120 

11880 

97150 

02850 

39 

22 

76958 

23042 

88287 

11713 

97285 

02715 

38 

23 

77173 

22827 

88453 

11547 

97421 

02579 

37 

24 

77387 

22613 

88618 

11382 

97556 

02444 

36 

25 

77600 

22400 

88783 

11217 

97691 

02309 

35 

26 

77811 

22189 

88948 

11052 

97825 

02175 

34 

27 

78022 

21978 

89111 

10889 

97959 

02041 

33 

28 

78232 

217C8 

89274 

10726 

98092 

01908 

3-2 

29 

78441 

21559 

89437 

10563 

98225 

01775 

31 

30 

8.78649 

11.21351 

8.89598 

11.10402 

8.98358 

11.01642 

30 

31 

78855 

21145 

89760 

10240 

98490 

01510 

29 

32 

79061 

20939 

89920 

10080 

98622 

01378 

28 

33  - 

79266 

20734 

90080 

09920 

9S753 

01247 

27 

34 

79470 

20530 

90240 

09760 

98884 

01116 

26 

35 

79673 

20327 

90399 

09601 

99015 

009S5 

25 

36 

79875 

20125 

90557 

09443 

99145 

00855 

24 

37 

80076 

19924 

90715 

09285 

99-J75 

00725 

23 

38 

80277 

19723 

90872 

09128 

99405 

00595 

22 

39 

80476 

19524 

91029 

08971 

99534 

00466 

21 

40 

8.80674 

11.19326 

8.91185 

11.08815 

8.99662 

11.00338 

20 

41 

80872 

191458 

91340 

08660 

99791 

00009 

19 

42 

81068 

18932 

91495 

08505 

99919 

00081 

18 

43 

81264 

18736 

91650 

08350 

9.00046 

10.99954 

17 

44 

81459 

18541 

91803 

08197 

00174 

99826 

16 

45 

81653 

18347 

91957 

OS043 

00301 

99699 

15 

46 

81  846 

18154 

9:2110 

07890 

00427 

99573 

14 

47 

82038 

17962 

92262 

07738 

00553 

99447 

18 

48 

82230 

17770 

92414 

07586 

00679 

9982] 

12 

49 

82420 

17580 

92565 

07435 

00805 

99195 

11 

50 

8.82610 

11.17390 

8.92716 

11.07284 

9.00930 

10.99070 

10 

51 

82799 

17201 

92866 

07134 

oio:>5 

98945 

9 

52 

82987 

17013 

93016 

06984 

01179 

98821 

8 

53 

83175 

16825 

93165 

06835 

01  303 

98697 

7 

5» 

83361 

16639 

93313 

06r~,87 

01427 

98673 

6 

55 

83547 

16453 

93462 

06538 

01550 

98450 

5 

56 

83732 

16268 

93609 

06391 

01673 

98327 

4 

57 

83916 

16084 

93756 

06244 

01796 

98204 

3 

58 

84100 

15900 

93903 

06097 

01918 

98082 

2 

59 

84282 

15718 

94049 

05951 

03040 

97960 

1 

60 

84464 

15536 

94195 

05805 

0216-2 

97838  !   0 

/ 

Cot  an 

Tan 

Cotan 

Tan 

Cotan 

Tan     , 

86° 

85° 

84° 

TABLE  Till.— LOG.  TANGENTS   AND  COTANGENTS.   255 


6 

o 

7° 

8° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.02162 

10.97838 

9.08914 

10.910S6 

9.14780 

10.85220 

60 

1 

02283 

97717 

09019 

90981 

14872 

85128 

59 

o 

02404 

97596 

09123 

90877 

14963 

85037 

58 

3 

02525 

97475 

09227 

90773 

15054 

84946 

57 

4 

02645 

97355 

09330 

90670 

15145 

84855 

56 

5 

02766 

97234 

09434 

90566 

15236 

84764 

55 

6 

02885 

97115 

09537 

90463 

15327 

84673 

54 

7 

03005 

96995 

09640 

90360 

15417 

84583 

53 

8 

03124 

96876 

09742 

90258 

15508 

84492 

52 

9 

03242 

96758 

09845 

90155 

15598 

84402 

51 

10 

9.03361 

10.96639 

9.09947 

10.90053 

9.15688 

10.84312 

50 

11 

03479 

96521 

10049 

89951 

15777 

84223 

49 

12 

03597 

96403 

10150 

89850 

15867 

84133 

48 

13 

03714 

96286 

10252 

89748 

15956 

84044 

47 

14 

03832 

96168 

10353 

89647 

16046 

83954 

46 

15 

03948 

96052 

10454 

89546 

16J35 

83865 

45 

16 

04065 

95935 

10555 

89445 

16224 

83776 

44 

17 

04181 

95819 

10656 

89344 

16312 

83688 

43 

18 

04297 

95703 

10756 

89244 

16401 

83599 

42 

19 

04413 

95587 

10856 

89144 

16489 

83511 

41 

20 

9.04528 

10.95472 

9.10956 

10.89044 

9.16577 

10.83423 

40 

21 

04643 

95357 

11058 

88944 

16665 

83335 

39 

22 

04758 

95242 

11155 

88845 

16753 

83247 

38 

23 

04873 

95127 

11254 

88746 

16841 

83159 

37 

24 

04987 

95013 

11353 

88647 

16928 

83072 

36 

25 

05101 

94899 

11452 

88548 

17016 

82984 

35 

26 

05214 

94786 

11551 

8S449 

17103 

82897 

34 

27 

05328 

94672 

11649 

88351 

17190 

82810 

33 

28 

05441 

94559 

11747 

88253 

17277 

82723 

32 

29 

05553 

94447 

11845 

88155 

17363 

82637 

31 

30 

9.05666 

10.94334 

9.11943 

10.88057 

9.17450 

10.82550 

30 

31 

0577'8 

94222 

1204C 

87960 

17536 

82464 

29 

32 

05890 

94110 

12138 

87862 

17622 

82378 

28 

33 

06002 

93998 

12235 

87765 

17708 

82292 

27 

34 

06113 

93887 

12332 

87668 

17794 

82206 

26 

35 

06224 

93776 

12428 

87572 

17880 

82120 

25 

36 

06335 

93(565 

12525 

87475 

17965 

82035 

24 

37 

06445 

93555 

12621 

87379 

18051 

81949 

23 

38 

06556 

93444 

12717 

87283 

18136 

81864 

22 

39 

06666 

93334 

12813 

87187 

18221 

81779 

21 

40 

9.06775 

10.93225 

9.12909 

10.87091 

9.18306 

10.81694 

20 

41 

06885 

93115 

13004 

86096 

18391 

81609 

19 

42 

06994 

93006 

13099 

8(5901 

1847'5 

81525 

18 

43 

07103 

92897 

13194 

86806 

18560 

81440 

17 

44 

07211 

92789 

13289 

86711 

18644 

81356 

16 

45 

07320 

92680 

K53S4 

8G616 

187'28 

81272 

15 

46 

07428 

92572 

13478 

86522 

18812 

81188 

14 

47 

07536 

92404 

13573 

86427 

18806 

81104 

13 

48 

07643 

9-:357 

13667 

863S3 

18979 

81021 

1\! 

49 

07751 

92249 

13761 

86239 

19063 

80937 

11 

50 

9.07858 

10.92142 

9.13854 

10.86146 

9.19146 

10.80854 

10 

51 

07964 

92036 

13948 

86052 

19229 

80771 

9 

52 

08071 

91929 

14041 

85959 

19312 

80688 

8 

53 

08177 

91823 

14134 

85866 

19395 

80605 

7 

54 

082*3 

91717 

14227 

85773 

19478 

80522 

6 

55 

08389 

91611 

14320 

85680 

19561 

80439 

5 

56 

08495 

91505 

14412 

85538 

19643 

80357 

4 

57 

08600 

91400 

14504 

85496 

19725 

80275 

3 

58 

08705 

91295 

14597 

85403 

19807 

80193 

2 

59 

08810 

91190 

14688 

85312 

19889 

80111 

1 

60 

OS914 

91086 

i  ;;so 

85220 

19971 

80029 

0 

, 

Cotan 

Tan 

Col.  "it 

Tan 

Cotan 

Tan 

/ 

83° 

82° 

81° 

256  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


9° 

10° 

11° 

| 

Tan 

Cotan 

Tan 

Cotau 

Tan 

Cotan 

0 

9.19971 

10.80029 

9.24632 

10.75368 

9.2S865 

10.71135 

60 

1 

20053 

79947 

24706 

75294 

2S983 

71C67 

59 

2 

20134 

79866 

21779 

75221 

29000 

71000 

58 

3 

20216 

79784 

24853 

75147 

29067 

70933 

57 

4 

20297 

79703 

24926 

75074 

29134 

70866 

56 

5 

20378 

79622 

25000 

75000 

29201 

70799 

55 

6 

20459 

79541 

25073 

74927 

29268 

70732 

54 

7 

20540 

79460 

25146 

74854 

29335 

70665 

53 

8 

20621 

79379 

25219 

74781 

29402 

70598 

52 

9 

20701 

79299 

25292 

74708 

29468 

70532 

51 

10 

9.20782 

10.79218 

9.25365 

10.746% 

9.29535 

10.70465 

50 

11 

20862 

79138 

25437 

74563 

29601 

70399 

49 

12 

20942 

79058 

25510 

74490 

29668 

70332 

48 

13 

21022 

78978 

25582 

74418 

29734 

70266 

47 

14 

21102 

78898 

25655 

74345 

29800 

70:200 

46 

15 

21182 

78818 

25727 

74273 

29866 

70134 

45 

16 

21261 

78739 

25799 

74201 

29932 

70068 

44 

17 

21341 

78659 

25871 

74129 

29998 

70002 

43 

18 

21420 

78580 

25943 

74057 

30064 

69930 

42 

19 

21499 

78501 

26015 

73985 

30130 

69870 

41 

20 

9.21578 

10.78422 

9.26086 

10.73914 

9.30195 

10.69805 

40 

21 

21657 

78343 

26158 

73842 

30261 

69739 

39 

22 

21736 

78264 

26229 

73771 

30326 

69674 

38 

23 

21814 

78186 

26301 

73699 

30391 

09609 

37 

24 

21893 

78107 

20372 

73628 

30457 

69543 

36 

25 

21971 

78029 

26443 

73557 

30522 

69478 

35 

26 

22049 

77951 

2(5514 

73486 

30587 

69413 

34 

27 

22127 

77873 

26585 

73415 

30652 

69348 

33 

28 

22205 

77795 

26655 

73345 

30717 

69283 

• 

29 

22283 

77717 

26726 

78274 

30782 

69218 

31 

30 

9.22361 

10.77639 

9.26797 

10.73203 

9.30S46 

10.69154 

30 

31 

22438 

77562 

26kl67 

73133 

30911 

69089 

29 

32 

22516 

77484 

26937 

73063 

30975 

690-35 

28 

33 

22598 

77407 

27008 

72992 

31040 

68960 

27 

34 

22670 

77330 

27078 

72922 

31104 

68896 

26 

35 

22747 

77253 

27148 

72852 

31168 

68832 

25 

36 

22824 

77176 

27218 

72782 

31233 

68767 

24 

37 

22901 

77099 

27288 

72712 

31297 

68703 

23 

38 

22977 

77023 

27357 

72643 

31361 

68639 

22 

39 

23054 

76946 

27427 

72573 

31425 

6S575 

21 

40 

9.23130 

10.76870 

9.27496 

10.72504 

9.31489 

10.68511 

m 

41 

28206 

76794 

27566 

72434 

31552 

68448 

19 

42 

28283 

76717 

27635 

72365 

31616 

68384 

18 

43 

23359 

76641 

27704 

72296 

31079 

68821 

17 

44 

23435 

76565 

27773 

72887 

3  1  743 

68257 

16 

45 

23510 

76490 

27842 

72158 

31806 

6?lfl4 

15 

46 

23586 

76414 

27911 

72089 

31870 

68130 

14 

47 

23661 

76339 

27980 

72020 

31933 

88067 

13 

48 

23737 

76263 

28049 

71951 

31  9H6 

6*004 

12 

49 

23312 

761H8 

28117 

71883 

32059 

67941 

11 

50 

9.23887 

10.76113 

9.2S186 

10.71814 

9.32122 

10.67KX 

10 

51 

2896-3 

76038 

28264 

71746 

32185 

67*15 

| 

H 

24037 

75968 

28323 

71677 

32248 

67752 

8 

53 

24112 

75888 

28391 

71(109 

3--'31  1 

676S9 

7 

54 

21180 

75814 

28459 

71541 

32373 

67627 

6 

55 

24201 

75739 

28527 

71473 

32436 

67504 

5 

56 

24335 

75665 

28595 

71405 

32498 

67502 

4 

57 

24410 

75590 

28662 

71338 

32561 

07439 

3 

58 

24484 

75516 

28730 

71270 

3'.)623 

67277 

5> 

59 

21558 

75442 

287»8 

71202 

32085 

67315 

1 

60 

24632 

75368 

28865 

71135 

32747 

87253 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Co  ran 

Tan 

80" 

7!» 

78° 

TABLE  VIII.—  LOG.  TANGENTS   AND    COTANGENTS.    257 


/ 

12° 

13° 

14° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.32747 

10.67253 

9.36336 

10.63664 

9.39677 

10.60323 

60 

1 

32810 

67190 

36394 

63606 

39731 

60269 

59 

2 

32872 

67128 

36452 

63548 

39785 

60215 

58 

3 

32933 

67067 

36509 

63491 

39838 

60162 

57 

4 

32995 

67005 

36566 

63434 

89892 

60108 

56 

5 

33057 

66943 

36624 

63376 

39945 

60055 

55 

6 

33119 

66881 

36681 

63319 

39999 

60001 

54 

7 

83180 

66820 

36738 

6326.2 

40052 

59948 

53 

8 

33v!42 

66758 

36795 

63205 

40106 

59894 

52 

9 

33303 

66697 

36852 

63148 

40159 

59841 

51 

10 

9.33365 

10.66635 

9.36909 

10.63091 

9.40212 

10.59788 

50 

11 

33426 

66574 

36966 

63034 

40266 

59734 

49 

12 

33487 

66513 

37023 

62977 

40319 

59681 

48 

13 

33548 

66452 

37080 

62920 

40372 

59628 

47 

14 

33609 

66391 

37137 

62863 

40425 

59575 

46 

15 

33670 

66330 

37193 

62807 

40478 

59522 

45 

16 

33731 

66269 

37250 

62750 

40531 

59469 

44 

17 

33792 

66208 

37306 

62694 

40584 

59416 

43 

18 

33853 

66147 

37363 

62637 

40636 

59364 

42 

19 

33913 

66087 

37419 

62581 

40689 

59311 

41 

20 

9.33974 

10.66026 

9.37476 

10.62524 

9.40742 

10.59258 

40 

21 

34034 

65966 

37532 

62468 

40795 

59205 

39 

22 

34095 

65905 

37588 

62412 

40847 

59153 

38 

23 

34155 

65845 

37644 

62356 

40900 

59100 

37 

24 

34215 

66785 

37700 

62300 

40952 

59048 

36 

25 

34276 

65724 

37756 

62244 

41005 

58995 

35 

26 

34336 

65664 

37812 

62188 

41057 

58943 

34 

27 

34396 

65604 

37868 

62132 

41109 

58891 

33 

28 

34456 

65544 

37924 

62076 

41161 

58839 

32 

29 

34516 

65484 

37980 

62020 

41214 

58786 

31 

30 

9.34576 

10.65424 

9.38035 

10.61965 

9.41266 

10.58734 

30 

n 

34035 

65365 

38091 

61909 

41318 

58682 

29 

32 

34695 

65305 

38147 

61853 

41370 

58630 

28 

33 

34755 

65245 

38202 

61798 

41422 

58578 

27 

34 

34814 

65186 

38257 

61743 

41474 

58526 

26 

85 

34874 

65126 

38313 

61687 

41526 

58474 

25 

36 

34933 

65067 

38368 

61632 

41578 

58422 

24 

37 

34992 

65008 

38423 

61577 

41629 

58371 

23 

38 

35051 

64949 

38479 

61521 

41681 

58319 

22 

39 

35111 

64889 

38534 

61466 

41733 

58267 

21 

40 

9.35170 

10.64830 

9.38589 

10.61411 

9.41784 

10.58216 

20 

41 

35229 

64771 

38844 

61356 

41836 

58164 

19 

42 

35288 

64712 

38699 

61301 

41887 

58113 

18 

43 

35347 

64663 

38754 

61246 

41939 

58061 

17 

44 

35405 

64595 

3S808 

61192 

41990 

58010 

16 

45 

35164 

645-3(5 

38S63 

61137 

42041 

57959 

15 

46 

35523 

64477 

38918 

61082 

42093 

57907 

14 

47 

35581 

64419 

38972 

61028 

42144 

57856 

13 

48 

35(540 

64360 

39027 

60973 

42195 

57805 

12 

49 

35098 

64302 

39082 

60918 

42246 

57754 

11 

50 

9.35757 

10.64243 

9.39136 

10.60864 

9.42297 

10.57703 

10 

51 

35815 

64185 

39190 

60810 

42348 

57652 

9 

52 

35873 

64127 

39245 

60755 

42399 

57601 

8 

53 

35931 

64069 

39299 

60701 

42450 

57550 

7 

54 

35989 

64011 

39853 

60647 

42501 

57499 

6 

55 

36047 

63953 

89407 

60593 

42552 

57448 

5 

56 

36105 

63895 

39461 

60539 

42603 

57397 

4 

57 

36163 

63837 

39515 

60485 

42653 

57347 

3 

58 

36221 

63779 

39569 

60431 

42704 

57296 

2 

59 

36279 

63721 

39623 

60377 

42755 

57245 

1 

60 

36336 

63664 

39677 

60323 

42805 

57195 

0 

/ 

Cotau 

Tan 

Cotan 

Tau 

Cotan 

Tau 

77° 

76° 

75° 

258    TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


f 

15° 

16° 

17° 

Tan 

Co  tan 

Tan 

Cotan 

Tan 

Col  an 

0 

9.42805 

10.57195 

9.45750 

10.54250 

9.48534 

10.51466 

60 

1 

42856 

57144 

45797 

54203 

48579 

51421 

59 

2 

42906 

57094 

45845 

54155 

48624 

51376 

58 

3 

42957 

57043 

45892 

54108 

48669 

51331 

57 

4 

43007 

56993 

45940 

54060 

48714 

51286 

56 

5 

43057 

56943 

45987 

54013 

48759 

51241 

55 

6 

43108 

56892 

46035 

53965 

48804 

51196 

54 

7 

43158 

56842 

46082 

53918 

48849 

51151 

53 

8 

43208 

56792 

46130 

53870 

48894 

51106 

52 

9 

43258 

56742 

46177 

53823 

48939 

51061 

51 

10 

9.43308 

10.56692 

9.46224 

10.53776 

9.48984 

10.51016 

50 

11 

43358 

56642 

46271 

53729 

49029 

50971 

49 

12 

43408 

56592 

46319 

53681 

49073 

50927 

48 

13 

43458 

56542 

46366 

53634 

49118 

50882 

47 

14 

43508 

56492 

46413 

53587 

49163 

50H37 

46 

15 

43558 

56442 

46460 

53540 

41)207 

50793  . 

45 

16 

43607 

56393 

46507 

53493 

49252 

50748 

44 

17 

43657 

56343 

46554 

53446 

49296 

50704 

43 

18 

43707 

56293 

46601 

53399 

49341 

50659 

42 

19 

43756 

56244 

46648 

53352 

49385 

50615 

41 

20 

9.43806 

10.56194 

9.46694 

10.53306 

9.49430 

10.50570 

40 

21 

43855 

56145 

46741 

53259 

49474 

50526 

39 

22 

43905 

56095 

46788 

53212 

49519 

50481 

38 

23 

43954 

56046 

46835 

53165 

49563 

50437 

37 

24 

44004 

55996 

46881 

53119 

49607 

50393 

36 

25 

44053 

55947 

46928 

53072 

49652 

50348 

35 

26 

44102 

55898 

46975 

53025 

49696 

50304 

34 

27 

44151 

55849 

47021 

52979 

49740 

50260 

33 

28 

44201 

55799 

47068 

52932 

49784 

50216 

32 

29 

44250 

55750 

47114 

52886 

49828 

50172 

31 

30 

9.44299 

10.55701 

9.47160 

10.52840 

9.49872 

10.50128 

30 

31 

44348 

55652 

47207 

52733 

49916 

50084 

29 

32 

44397 

55603 

47253 

52747 

49960 

50040 

28 

33 

44446 

55554 

47299 

52701 

50004 

49996 

27 

34 

44495 

55505 

47346 

52654 

50048 

49952 

26 

35 

44544 

55456 

47392 

52608 

50092 

49908 

25 

36 

44592 

55408 

47438 

52562 

50136 

49864 

24 

37 

44641 

55359 

47484 

52516 

50180 

49820 

23 

38 

44690 

55310 

47530 

52470 

50J23 

49777 

22 

39 

44738 

55262 

47576 

52424 

50267 

49733 

21 

40 

9.44787 

10.55213 

9.47622 

10.52378 

9.50311 

10.49689 

20 

41 

44836 

55164 

47668 

52332 

50355 

49645 

19 

42 

44884 

55116 

47714 

52286 

50398 

49602 

18 

43 

44933 

55067 

47760 

52240 

50442 

49558 

17 

44 

44981 

55019 

47806 

52194 

50485 

49515 

16 

45 

45029 

54971 

47852 

52148 

50529 

49471 

15 

46 

45078 

54922 

47897 

52103 

50572 

49428 

14 

47 

4512(5 

54874 

47943 

52057 

50616 

493C4 

13 

48 

45174 

54820 

479S9 

52011 

50659 

49341 

12 

49 

45222 

54778 

48035 

51965 

50703 

49297 

11 

50 

9.45271 

10.54729 

9.48080 

10.51920 

9.50746 

10.49254 

10 

51 

45319 

54681 

48126 

51874 

50789 

49211 

9 

52 

45367 

54633 

48171 

51829 

50833 

49167 

8 

53 

45415 

54585 

4S217 

51783 

50876 

49124 

r* 

54 

45463 

54537 

48262 

51738 

50919 

49081 

6 

5.r> 

45511 

54489 

48307 

51693 

50962 

49038 

5 

56 

45559 

54441 

48353 

51647 

51005 

4S<j(.r> 

4 

57 

45606 

54394 

48398 

51602 

51048 

48952 

3 

58 

45654 

54:346 

48443 

51557 

51092 

48908 

2 

59 

45702 

54298 

484H9 

51511 

51135 

48865 

1 

60 

45750 

54250 

48534 

5146C 

51178 

48822 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan    , 

74° 

73° 

72° 

TABLE  VIII.— LOG.  TANGENTS    AND   COTANGENTS.   259 


18° 

19° 

20° 

f 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.51178 

10  48822 

9.53697 

10.46303 

9.56107 

10.43893 

00 

1 

51221 

48779 

53738 

46262 

56146 

43S54 

59 

<> 

51204 

48736 

53779 

46221 

56185 

438.5 

58 

3 

51306 

48694 

538*0 

46180 

56224 

43776 

57 

4 

51349 

48651 

53861 

46189 

56264 

43736 

56 

5 

51893 

48608 

53902 

46098 

56303 

43697 

55 

6 

51435 

485C5 

53943 

46057 

56342 

43658 

54 

51478 

48522 

53984 

46016 

56381 

43619 

53 

8 

51520 

48480 

54025 

45975 

56420 

43580 

52 

9 

51563 

48437 

54065 

45935 

56459 

43541 

51 

10 

9.51606 

10.48394 

9.54106 

10.45894 

9.56498 

10.43502 

50 

11 

51648 

48352 

54147 

45853 

56537 

43463 

49 

12 

51691 

48309 

54187 

45813 

56576 

43424 

48 

13 

51734 

48266 

54228 

45772 

56815 

43385 

47 

14 

51776 

48224 

54269 

45731 

56654 

43346 

46 

15 

51819 

48181 

54309 

45691 

56693 

43307 

45 

16 

51861 

48139 

54350 

45650 

56733 

43268 

44 

17 

51903 

48097 

54390 

45610 

5677  1 

43329 

43 

18 

51946 

48054 

54431 

45569 

56810 

43190 

42 

19 

51988 

48012 

54471 

45529 

56849 

43151 

41 

20 

9.52031 

10.47969 

9.54512 

10.45488 

9.56887 

10.43113 

40 

21 

52073 

47927 

54552 

45448 

56926 

43074 

39 

22 

52115 

47885 

54593 

45407 

56965 

43035 

38 

33 

52157 

47843 

54033 

45367 

57004 

42996 

37 

24 

52200 

47800 

54673 

45327 

57042 

42958 

86 

25 

52242 

47758 

54714 

45286 

57081 

42919 

35 

26 

52284 

47716 

54754 

45246 

57120 

42880 

34 

52326 

47674 

54794 

45206 

57158 

42842 

33 

28 

52368 

47632 

54835 

45165 

57197 

42803 

32 

29 

52410 

47590 

54875 

45125 

57235 

42765 

31 

30 

9.52452 

10.47548 

9.54915 

10.45085 

9.57274 

10.42726 

30 

31 

52494 

47506 

54955 

45045 

57312 

42688 

29 

32 

52536 

47464 

54995 

45005 

57351 

42649 

28 

33 

52578 

47422 

55035 

44965 

57389 

42611 

27 

34 

52620 

47380 

55075 

44925 

57428 

42572 

26 

35 

52661 

47339 

55115 

44885 

57466 

42534 

25 

36 

52708 

47297 

55155 

44845 

57504 

42496 

24 

37 

52745 

47255 

55195 

44805 

57543 

42457 

23 

38 

52787 

47213 

55235 

44765 

57581 

42419 

22 

39 

52829 

47171 

55275 

44785 

57619 

42381 

21 

40 

9.52870 

10.47130 

9.55315 

10.44685 

9.57658 

10.42342 

20 

41 

52912 

47088 

55355 

44645 

57696 

42304 

19 

42 

52953 

47047 

55395 

44605 

57734 

42266 

18 

43 

52995 

47005 

55434 

44566 

57772 

42228 

17 

44 

53037 

46963 

55474 

4J5'?6 

57810 

42190 

16 

45 

53078 

4(5932 

55514 

44486 

57849 

42151 

15 

46 

53120 

46880 

55554 

44446 

57887 

42113 

14 

47 

53161 

46839 

55593 

44407 

57925 

42075 

13 

48 

53202 

46798 

55633 

448(57 

57908 

42037 

12 

49 

53244 

46756 

55673 

44327 

58001 

41999 

11 

50 

9.53285 

10.46715 

9.55712 

10.44288 

9.58039 

10.41961 

10 

51 

533-27 

46673 

55752 

44248 

58077 

41923 

9 

52 

53308 

46G32 

55791 

44209 

58115 

41885 

8 

53 

53409 

46591 

55831 

44169 

58153 

41847 

y 

54 

53450 

46550 

55870 

44130 

58191 

41809 

6 

55 

53492 

46508 

55910 

44090 

58229 

41771 

5 

56 

53533 

46467 

55949 

44051 

58267 

41733 

4 

57 

53574 

46426 

55989 

44011 

58304 

41696 

3 

58 

53615 

46385 

56028 

43972 

58342 

41658 

2 

59 

53656 

46344 

56067 

43933 

58380 

41620 

1 

60 

53697 

46303 

56107 

43893 

56418 

41582 

0 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

71° 

70° 

09°      | 

260  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

21o 

22° 

23° 

Tan 

Cotan 

Tan 

Cotau 

Tan 

Cotan 

0 

9.58418 

10.41582 

9.60641 

10.39359 

9.62785 

10.37215 

60 

1 

58155 

41545 

60677 

39323 

62820 

37180 

59 

2 

58493 

41507 

60714 

39286 

62855 

37145 

58 

3 

58531 

41469 

60750 

39250 

62890 

37110 

57 

4 

58509 

41431 

60786 

39214 

62926 

37074 

56 

5 

58006 

41394 

60823 

39177 

62961 

37039 

55 

6 

58644 

41356 

60859 

39141 

62996 

37004 

54 

58081 

41319 

60895 

39105 

63031 

30969 

53 

8 

58719 

41281 

60931 

39069 

63066 

30934 

52 

9 

58757 

41243 

60967 

39033 

63101 

36899 

51 

10 

9.58794 

10.41206 

9.61004 

10.38996 

9.63135 

10.36865 

50 

11 

5883-3 

41168 

61040 

38960 

63170 

36830 

49 

12 

58869 

41131 

61076 

38924 

63205 

36795 

48 

13 

58907 

41093 

61112 

38888 

63240 

36760 

47 

14 

58944 

41056 

61148 

38852 

63275 

36725 

46 

15 

58981 

41019 

61184 

38816 

63310 

30690 

45 

16 

59019 

40981 

61220 

38780 

63345 

36655 

44 

17 

59056 

401)44 

61256 

38744 

63379 

30021 

43 

18 

59094 

40906 

61292 

38708 

63414 

36586 

42 

19 

59131 

40869 

61328 

38672 

63449 

36551 

41 

20 

9.59168 

10.40832 

9.61304 

10.38636 

9.63484 

10.36516 

40 

21 

59-205 

40795 

61400 

38(500 

63519 

36481 

39 

22 

59243 

40757 

61430 

38564 

63553 

36447 

38 

23 

59380 

40720 

61472 

88528 

63588 

36412 

37 

24 

59317 

40(583 

61508 

3*492 

63623 

30377 

36 

25 

59354 

40646 

61544 

38456 

63(557 

36313 

35 

26 

59391 

40609 

61579 

38421 

03092 

36308 

34 

27 

59429 

40571 

61615 

38385 

03726 

3(5274 

33 

28 

594  06 

40534 

61651 

383  19 

63701 

30219 

32 

29 

59503 

40497 

61687 

38313 

63796 

36204 

31 

30 

9.59540 

10.40460 

9.61722 

10.38-278 

9.63830 

10.36170 

30 

31 

59577 

40423 

61758 

88242 

63865 

36135 

2!) 

82 

59614 

40386 

61794 

38206 

63899 

36101 

28 

33 

59651 

40349 

61830 

38170 

(53934 

3(5006 

27 

§4 

59688 

40312 

61S65 

3S135 

63968 

30032 

86 

85 

59725 

40275 

61901 

3H099 

f.4003 

35997 

25 

36 

59762 

40238 

61930 

3S004 

64037 

35903 

24 

37 

59799 

40201 

61972 

38028 

64072 

35928 

23 

38 

59S35 

40165 

62008 

37992 

64106 

35894 

22 

85 

598^2 

40128 

62043 

37957 

64140 

35800 

21 

40 

9.59909 

10.40091 

9.6-2079 

10  37921 

9.64175 

10.358-25 

20 

41 

59946 

40054 

62114 

87886 

6421  '9 

35791 

19 

42 

59983 

40017 

62150 

37850 

64243 

85757 

18 

43 

60019 

39D81 

6-2185 

87815 

64278 

35722 

17 

44 

00056 

399  14 

68831 

37779 

64312 

35688 

1(5 

45 

60093 

391)117 

62256 

37744 

64346 

35654 

15 

40 

60130 

39870 

0-2-292 

37708 

64381 

35619 

14 

47 

601(16 

89684 

0-23-27 

87678 

64415 

35585 

13 

48 

60203 

89797 

6-2302 

37638 

01449 

35551 

12 

49 

60240 

39760 

62398 

37602 

64483 

35517 

11 

50 

9.G0276 

10.39724 

9.62433 

10.37567 

9.64517 

10.35483 

10 

51 

00813 

39087 

62468 

37532 

64552 

35  MS 

•I 

52 

6034!) 

31)051 

62504 

37496 

64586 

35414 

8 

53 

60386 

39(514 

62539 

37401 

640-20 

85880 

r* 

54 

604-22 

30578 

62574 

37426 

64654 

85346 

6 

55 

60159 

31)511 

62009 

373U1 

040S8 

35312 

5 

56 

fi04!>5 

39505 

0-2015 

37355 

64722 

3.V27S 

4 

57 

80582 

89468 

62680 

373-20 

64756 

3524  1 

3 

68 

80568 

39  132 

62715 

87285 

(54190 

85210 

59 

<;o6or> 

3SI3!)5 

62750 

37250 

64824 

35176 

I 

60 

III  Kill 

39359 

62785 

87215 

61858 

85142 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

<  'oian 

Tan 

/ 

68° 

07° 

(Hi" 

TABLE  Vlit.— LOG.  TANGENTS   AND   COTANGENTS.    261 


/ 

24° 

25° 

2< 

J° 

f 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.64858 

10.35142 

9.66867 

10.33133 

9.68818 

10.31182 

60 

1 

64892 

35108 

66900 

33100 

68850 

31150 

59 

0 

64926 

35074 

66933 

33067 

68882 

31118 

58 

3 

64960 

35040 

66966 

33034 

68914 

31086 

57 

4 

64994 

35006 

66999 

33001 

68946 

31054 

56 

5 

65028 

84972 

67032 

32968 

68978 

31022 

55 

6 

650G2 

34938 

67065 

32935 

69010 

31990 

54 

7 

65096 

34904 

67098 

82902 

69042 

31958 

53 

8 

65130 

3-1870 

67131 

32869 

69074 

31926 

52 

9 

65164 

34836 

67163 

32837 

69106 

31894 

51 

10 

9.65197 

10.34803 

9.67196 

10.32804 

9.69138 

10.30862 

50 

11 

05281 

34769 

67229 

32771 

69170 

30830 

49 

12 

65265 

34735 

67262 

32738 

69202 

80798 

48 

13 

65299 

34701 

67295 

32705 

69234 

30766 

47 

14 

65333 

34667 

67327 

32673 

69266 

30734 

46 

15 

65366 

34634 

67360 

32640 

69298 

30702 

45 

16 

65400 

34600 

67393 

32007 

69329 

30671 

44 

17 

65484 

34566 

67426 

32574 

69361 

30639 

43 

18 

65467 

34533 

67458 

32542 

69393 

30607 

42 

19 

65501 

34499 

67491 

32509 

69425 

30575 

41 

20 

9.65535 

10.34465 

9.67524 

10.3247'6 

9.69457 

10.30543 

40 

21 

65508 

34432 

67556 

32444 

69488 

30512 

39 

22 

65002 

34398 

67589 

32411 

69520 

30480 

38 

23 

65636 

34364 

67622 

32378 

69552 

30448 

37 

24 

65609 

34331 

67654 

32346 

69584 

30416 

36 

25 

65703 

34297 

67687 

32313 

69615 

30385 

35 

26 

65736 

34264 

67719 

32281 

69647 

30353 

34 

27 

65770 

34230 

67752 

30248 

69679 

30321 

33 

28 

65S03 

34197 

67785 

32215 

69710 

30290 

32 

29 

65837 

34163 

67817 

321  S3 

69742 

30258 

31 

30 

9.65870 

10.34130 

9.67850 

10.32150 

9.69774 

10.30226 

30 

31 

65904 

34096 

67882 

32118 

09805 

30195 

29 

32 

65937 

34063 

67915 

32085 

69837 

30163 

28 

33 

65971 

34029 

67947 

32053 

69868 

30132 

27 

34 

66004 

33996 

67980 

32020 

69900 

30100 

26 

85 

66038 

33962 

68012 

31988 

69932 

30068 

25 

36 

66071 

33929 

68044 

31956 

69963 

30037 

24 

37 

66104 

33896 

68077 

31923 

69995 

30005 

23 

38 

66138 

33862 

68109 

31891 

70026 

29974 

22 

39 

60171 

33829 

68142 

31858 

70058 

29942 

21 

40 

9.66204 

10.33796 

9.68174 

10.31826 

9.70089 

10.29911 

20 

41 

66238 

33762 

68206 

31794 

70121 

29879 

19 

42 

66271 

33729 

68239 

31761 

70152 

29848 

18 

43 

66304 

88696 

68271 

31729 

70184 

29816 

17 

44 

66337 

33663 

68303 

31697 

70215 

29785 

16 

45 

«<537i 

33629 

68336 

31664 

70247 

29753 

15 

40 

66404 

835!K> 

68368 

31632 

70278 

29722 

14 

47 

66437 

33563 

68400 

31600 

70309 

29091 

13 

48 

56470 

33530 

68432 

31568 

70341 

29059 

12 

49 

60603 

33497 

68465 

31535 

70372 

29628 

11 

50 

8.66537 

10.33463 

9.68497 

10.31503 

9.70404 

10.295% 

10 

51 

66570 

33430 

68529 

31471 

70435 

29565 

9 

52 

66603 

33397 

68561 

31439 

70466 

29534 

8 

r,3 

66636 

33364 

68593 

31407 

70498 

29502 

7 

54 

66669 

33331 

6S020 

31374 

70529 

29471 

6 

55 

66702 

33298 

68658 

31342 

70560 

29440 

5 

56 

66785 

33265 

6S690 

31310 

70592 

29408 

4 

57 

60768 

33232 

68722 

81278 

70623 

29377 

3 

58 

66801 

33199 

68754 

31216 

70654 

29346 

2 

59 

66834 

33160 

68786 

31214 

70685 

29315 

1 

60 

66867 

33133 

68818 

31182 

70717 

29283 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

65° 

64° 

83° 

TABLE  VIII.— LOG.   TANGENTS   AND   COTANGENTS. 


0 

27°             28° 

29° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.70717 

10.29283 

9.72567 

10.27433 

9.74375 

10.25025 

00 

1 

70748 

29252 

72598 

27402 

74405 

25595 

59 

2 

70779 

29221 

72628 

27372 

74135 

25505 

58 

3 

70810 

29190 

72659 

27341 

74465 

25535 

57 

4 

70841 

29159 

72689 

27311 

74494 

25506 

56 

5 

70873 

29127 

72720 

27280 

74524 

25416 

55 

6 

70904 

29096 

72750 

27250 

74554 

25446 

54 

7* 

70935 

29065 

72780 

27220 

74583 

25417 

53 

8 

70966 

29034 

72811 

27189 

74613 

25387 

52 

9 

70997 

29003 

72841 

27159 

74643 

25357 

51 

10 

9.71028 

10.28972 

9.72872 

10.27128 

9.71073 

10.25327 

50 

11 

71059 

28941 

72902 

27098 

74702 

25298 

49 

12 

71090 

28910 

72932 

27068 

74732 

25268 

4S 

13 

71121 

28879 

72963 

27037 

74162 

25238 

47 

14 

71153 

28847 

12993 

21007 

74791 

25209 

46 

15 

71184 

28816 

73023 

26977 

74821 

25179 

45 

16 

71215 

28785 

73054 

26946 

74851 

25149 

44 

17 

71246 

28754 

73084 

26916 

74880 

25120 

43 

18 

71277 

28723 

73114 

26886 

74910 

25090 

42 

19 

71308 

28692 

73144 

26856 

74939 

25061 

41 

20 

9.71339 

10.28661 

9.73175 

10.26825 

9.74969 

10.25031 

40 

21 

71370 

28630 

13205 

26795 

74998 

25002 

39 

22 

71401 

28599 

73235 

26765 

75028 

24972 

38 

23 

71431 

28569 

73265 

26735  • 

75058 

24942 

37 

24 

71462 

28538 

73295 

26705 

75087 

24913 

36 

25 

71493 

28507 

73326 

26074 

75117 

24883 

35 

26 

71524 

28476 

73356 

26644 

75146 

24854 

34 

27 

71555 

28445 

73386 

26614 

75176 

24824 

33 

28 

71586 

28414 

13416 

26584 

75205 

24795 

32 

29 

71617 

28383 

73446 

26554 

75235 

24765 

31 

30 

9.71648 

10.28352 

9.73476 

10.26524 

9.75264 

10.24736 

30 

31 

71679 

28321 

73507 

26493 

75294 

24706 

29 

32 

71709 

28291 

73537 

26463 

75323 

24677 

28 

33 

71740 

28260 

73567 

26433 

75353 

24647 

27 

34 

71771 

28229 

13597 

26403 

75382 

24618 

26 

35 

71802 

28198 

13027 

26373 

75411 

24589 

25 

36 

71833 

28167 

73657 

26343 

75441 

24559 

24 

37 

71863 

28137 

73687 

26313 

75470 

24530 

23 

38 

71894 

28103 

73717 

26283 

75500 

24500 

22 

39 

71925 

28075 

73747 

26253 

75529 

24471 

21 

40 

9.71955 

10.28045 

9.73777 

10.26223 

9.75558 

10.24442 

20 

41 

71986 

28014 

73807 

26193 

75588 

24412 

19 

42 

72017 

27983 

73837 

26163 

75617 

24383 

18 

43 

72048 

27952 

73867 

26133 

75647 

24353 

17 

44 

72078 

27922 

73897 

26103 

75676 

24324 

16 

45 

72109 

27891 

73927 

20073 

75705 

24295 

15 

46 

72140 

27860 

73957 

26043 

75735 

24205 

14 

47 

72170 

27830 

73987 

26013 

75764 

24236 

13 

48 

72201 

27799 

74017 

25983 

75793 

24207 

12 

49 

72231 

27769 

74047 

25958 

75822 

24178 

11 

50 

9.72262 

10.27738 

9.74077 

10.25923 

9.75852 

10.24148 

10 

51 

72293 

27707 

74107 

25893 

75881 

24119 

<) 

52 

72323 

27677 

74137 

25863 

75910 

24090 

8 

53 

72354 

27646 

74166 

25834 

75939 

24061 

ij- 

54 

72384 

27616 

74196 

25804 

75969 

24031 

6 

55 

72415 

27585 

74226 

25774 

75998 

24002 

5 

56 

72445 

27555 

71256 

25744 

70027 

23973 

4 

57 

72476 

27524 

74886 

25714 

76056 

231144 

3 

58 

72506 

27494 

74316 

25084 

76086 

23914 

2 

59 

72537 

27463 

74345 

25655 

76115 

23885 

1 

60 

72567 

27433 

14375 

25625 

76144 

23856 

0 

, 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

02° 

01° 

60° 

TABLE  VIII.— LOG.  TANGENTS   AND  COTANGENTS.   263 


30°              31°              32° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.76144 

10.23856 

9.77877 

10.22123  ' 

9.79579 

10.20421 

60 

1 

76173 

23827 

77906 

22094 

79607 

20393 

59 

2 

76202 

23798 

77935 

22065 

79035 

20365 

58 

3 

76231 

23769 

77963 

22037 

79663 

20337 

57 

4 

76261 

23739 

77992 

22008 

79691 

20309 

56 

5 

76290 

23710 

78020 

21980 

79719 

20281 

55 

6 

76319 

23681 

78049 

21951 

79747 

20253 

54 

7 

76348 

23652 

78077 

21923 

79776 

20224 

53 

8 

76377 

23623 

78106 

21894 

79804 

20196 

52 

9 

76406 

23594 

78135 

21865 

79832 

20168 

51 

10 

9.76435 

10.23565 

9.78163 

10.21837 

9.79860 

10.20140 

50 

11 

764(54 

23536 

78192 

21808 

79888 

20112 

49 

12 

76493 

23507 

782-20 

21780 

79916 

20084 

48 

13 

76522 

23478 

78249 

21751 

79944 

20056 

47 

14 

76551 

23449 

78277 

21723 

79972 

20028 

46 

15 

76580 

23420 

78306 

21694 

80000 

20000 

45 

16 

76609 

23391 

78334 

21666 

80028 

19972 

44 

17 

76639 

23361 

78363 

21637 

80056 

19944 

43 

18 

76668 

23332 

78391 

21609 

80084 

19916 

42 

19 

76697 

23303 

78419 

21581 

80112 

19888 

41 

20 

9.76725 

10.23275 

9.78448 

10.21552 

9.80140 

10.19860 

40 

21 

76754 

23246 

78476 

21524 

80168 

19832 

39 

22 

76783 

23217 

78505 

21495 

80195 

19805 

38 

23 

76812 

23188 

78533 

21467 

80223 

19777 

37 

24 

76841 

23159 

78562 

21438 

80251 

19749 

36 

25 

76870 

23130 

78590 

21410 

80279 

19721 

35 

26 

76899 

23101 

78618 

21382 

80307 

19693 

34 

27 

76928 

28072 

78647 

21353 

80335 

19665 

33 

28 

76957 

23043 

78675 

21325 

80363 

19637 

32 

29 

76986 

23014 

78704 

21296 

80391 

19609 

31 

30 

9.77015 

10.22985 

9.78732 

10.21268 

9.80419 

10.19581 

30 

31 

77044 

22956 

78760 

21240 

80447 

19553 

29 

32 

77073 

22927 

78789 

21211 

80474 

19526 

28 

33 

77101 

22899 

78817 

21183 

80502 

19498 

27 

34 

77130 

22870 

78845 

21155 

80530 

19470 

26 

35 

77159 

22841 

78874 

21126 

80558 

19442 

25 

36 

77188 

22812 

78902 

21098 

80586 

19414 

24 

37 

77217 

22783 

78930 

21070 

80614 

19386 

23 

38 

77246 

22754 

78959 

21041 

80642 

19358 

22 

39 

77274 

22726 

78987 

21013 

80669 

19331 

21 

40 

9.77303 

10.22697 

9.79015 

10.20985 

9.80697 

10.19303 

20 

41 

77332 

22668 

79043 

20957 

80725 

19275 

19 

42 

77361 

22639 

79072 

20928 

80753 

19247 

18 

43 

77390 

22610 

79100 

20900 

80781 

19219 

17 

44 

77418 

22582 

79128 

20872 

80808 

19192 

16 

45 

77447 

22553 

79156 

20844 

80836 

19164 

15 

46 

77476 

22524 

79185 

20815 

80864 

19136 

14 

47 

77505 

22495 

79213 

20787 

80892 

19108 

13 

48 

77533 

22467 

79241 

20759 

80919 

19061 

12 

49 

77562 

22438 

79269 

20731 

80947 

19053 

11 

50 

9.77591 

10.22409 

9.79297 

10.20703 

9.80975 

10.19025 

10 

51 

77619 

22381 

79326 

20674 

81003 

18997 

9 

52 

77648 

22352 

79354 

20646 

81030 

18970 

8 

53 

77677 

22323 

79382 

20618 

81058 

18942 

54 

77706 

22294 

79410 

20590 

81086 

18914 

6 

55 

77734 

22266 

79438 

20562 

81113 

18887 

5 

56 

77763 

22237 

79466 

20534 

81141 

18859 

4 

57 

77791 

22209 

79495 

20505 

81169 

18831 

3 

58 

77820 

22180 

79523 

20477 

81196 

18804 

2 

59 

77849 

22151 

79551 

20449 

81224 

18776 

1 

60 

77877 

22123 

79579 

20421 

81252 

18748 

0 

, 

Cotau 

Tan 

Cotan 

Tan 

Cotan 

Tan 

i 

69° 

68° 

57° 

264  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


83° 

34° 

85° 

Tan 

Golan 

Tau 

Cotan 

Tan 

Cotan 

o 

9.81252 

10.18748 

9.82^99 

10.17101 

9.84523 

10.15477 

60 

I 

81279 

18r21 

82926 

17074 

81550 

15450 

59 

2 

81307 

18(593 

82953 

17047 

81576 

15424 

58 

J 

81335 

1866') 

829-0 

1  r020 

846L'3 

15397 

57 

4 

81  302 

18638 

8:5008 

10992 

84630 

15370 

56 

5 

81390 

18610 

88035 

16965 

84657 

15343 

55 

g 

81418 

18582 

83062 

1093S 

84684 

15316 

54 

81145 

18555 

83089 

Ki911 

84711 

16289 

53 

g 

81473 

18527 

83117 

16883 

84738 

15202 

52 

9 

81500 

18.500 

83144 

16856 

84764 

15236 

51 

10 

9.81528 

10.18472 

9.83171 

10.168-29 

9.84791 

10.15209 

50 

j  j 

81556 

18444 

83198 

16S02 

84818 

15182 

49 

If 

81583 

18417 

83225 

16775 

84845 

15155 

48 

18 

81611 

18389 

83252 

16748 

84872 

15128 

47 

14 

81638 

18863 

83280 

16720 

84899 

15101 

46 

15 

81666 

1833  J 

83307 

16693 

84925 

15075 

45 

16 

81693 

18307 

83334 

16666 

84952 

15048 

44 

17 

81721 

18279 

83361 

16639 

84979 

15021 

43 

1ft 

81748 

18252 

833S8 

16612 

85006 

14994 

42 

Jo 

19 

81776 

18224 

83415 

16585 

85033 

14967 

41 

20 

9.81803 

10.18197 

9.83142 

10.16558 

9.85059 

10.14941 

40 

21 

81831 

18169 

8:5470 

16530 

85086 

14914 

39 

22 

81858 

18142 

83497 

16503 

85113 

14887 

38 

23 

81886 

18114 

83524 

16476 

85140 

14860 

37 

24 

81913 

18087 

83551 

16449 

85166 

14834 

36 

25 

81941 

18059 

83578 

16422 

85193 

14807 

35 

26 

81968 

18032 

83605 

16395 

85220 

14780 

34 

27 

81996 

18004 

83682 

16368 

85247 

14753 

33 

OB 

82023 

17977 

83659 

10311 

85273 

14727 

32 

*o 

29 

82051 

17949 

83686 

16314 

85300 

14700 

31 

30 

9.82078 

10.17922 

9.83713 

10.16287 

9.85327 

10.14673 

30 

31 

82106 

17894 

83740 

16260 

86354 

14(140 

29 

32 

821  33 

178S7 

83768 

16232 

85380 

1  1620 

28 

33 

82161 

17839 

83795 

16-205 

85407 

14593 

27 

<JA 

82188 

17812 

83822 

16178 

85434 

14566 

26 

O-t 

35 

82215 

17785 

83849 

16151 

85460 

14540 

25 

oc 

82243 

17757 

83876 

16124 

85  1ST 

14513 

24 

oo 
37 

82270 

17730 

83903 

16097 

85514 

14480 

23 

OQ 

82298 

17702 

88988 

1607'0 

85540 

14460 

22 

OO 

39 

8232") 

17675 

83957 

1(5043 

85567 

14433 

21 

40 

9.82352 

10.17618 

9.83984 

10.16016 

9.85594 

10.14406 

20 

41 

82380 

17620 

84011 

15989 

85620 

14880 

19 

42 

82407 

17593 

84038 

15962 

85647 

14353 

18 

43 

82435 

17565 

84065 

15935 

85674 

14320 

17 

44 

82462 

17538 

84092 

15908 

85700 

14:500 

16 

45 

82489 

17511 

84119 

15881 

85727 

14273 

15 

46 

82517 

17483 

84  146 

16854 

85754 

14240 

14 

47 

82544 

17456 

84173 

15S27 

85780 

14220 

13 

48 

82571 

17429 

84200 

16800 

85807 

14193 

12 

49 

82599 

17401 

84227 

15773 

85884 

14166 

11 

50 

9.82626 

10.17374 

9.84254 

10.15746 

9.85800 

10.14140 

10 

82653 

17347 

8  I2SO 

15720 

85887 

14113 

«.) 

52 

82681 

17319 

84307 

15(193 

859  1  3 

14087           8 

53 

82708 

17292 

84334 

16666 

85940 

14060            7 

54 

82735 

172(55 

84361 

15639 

85967 

1  4033 

(i 

55 

82762 

1723S 

843S8 

1  5612 

N5<)93 

1  4007 

5 

56 

82790 

17210 

84415 

1  55V.-, 

8(1020 

18980 

4 

57 

82817 

17183 

84442 

15558 

86046 

18954 

3 

KQ 

82844 

17156 

84469 

15531 

8(1073 

18!  127 

2 

OO 

59 

82871 

17129 

84406 

15504 

861  (HI 

13900 

1 

60 

82899 

17101 

S4523 

15477 

86126 

18874 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

66° 

65° 

64° 

TABLE  VIII.— LOG.  TAN<!EXTS    AND   COTANGENTS.    265 


3 

0° 

87° 

38 

° 

f 

Tan 

Cotan 

Tan 

Cotan 

T;u. 

Cotan 

0 

9.86126 

10.13874 

9.87711 

10.12289 

9.89281 

10.10719 

60 

1 

86153 

13847 

87738 

12262 

89307 

10693 

59 

2 

86179 

13S21 

87764 

12236 

89333 

10C67 

58 

3 

86206 

13794 

87790 

12210 

89359 

10641 

57 

4 

86232 

13768 

87817 

12183 

89385 

10615 

56 

5 

86259 

13741 

87843 

12157 

89411 

10589 

55 

6 

86285 

13715 

87869 

12131 

89487 

10563 

54 

86312 

13688 

87895 

12105 

89463 

10537 

53 

8 

86338 

13662 

87922 

12078 

89489 

10511 

52 

9 

86365 

13635 

87948 

12052 

89515 

10485 

51 

10 

9.86392 

10.1360S 

9.87.974 

10.12026 

9.89541 

10.10459 

50 

11 

86418 

13582 

88000 

12000 

89567 

10433  i  49 

12 

86445 

13555 

88027 

11973 

89593 

10407    48 

13 

86471 

13559 

88053 

11947 

89619 

10381 

47 

14 

86498 

13502 

88079 

11921 

89645 

10355 

46 

15 

86524 

13476 

88105 

11895 

89(571 

10329 

45 

16 

86551 

13449 

88131 

11869 

89697 

10303 

44 

17 

86577 

13423 

88158 

11842 

89723 

10277 

43 

18 

86603 

13397 

88184 

11816 

89749 

10251 

42 

19 

86630 

13370 

88210 

11790 

89775 

10225 

41 

20 

9.86656 

10.13344 

9.88236 

10.11764 

9.89801 

10.10199 

40 

21 

86ti83 

13317 

882C2 

11738 

89827 

10173 

39 

22 

86709 

13291 

88289 

11711 

89853 

10147 

38 

23 

86J36 

13264 

88315 

11685 

89879 

10)21 

37 

24 

86762 

13238 

88341 

11659 

89905 

10095 

36 

25 

86789 

13211 

8N367 

11633 

89931 

10069 

35 

26 

86815 

13185 

88893 

11(507 

89957 

10043 

34 

27 

86842 

13158 

88420 

11580 

89983 

10017 

33 

28 

86868 

13132 

88446 

11554 

90009 

C99!)  1 

32 

29 

86894 

13106 

88472 

11528 

1)0035 

09965 

31 

30 

9.86921 

10.13079 

9.88498 

10.11502 

9.90081 

10.09939 

30 

31 

86947 

13058 

88524 

11476 

90086 

09914 

29 

32 

86974 

13026 

88550 

11450 

90112 

09888 

28 

33 

87000 

13000 

88577 

11423 

90138 

09862 

27 

34 

87027 

12973 

88603 

11397 

90164 

09836 

26 

35 

87053 

12947 

88629 

11371 

90190 

09H10 

25 

36 

87079 

12921 

88655 

11345 

90216 

09784 

24 

37 

87106 

12894 

88681 

11319 

90242 

09758 

23 

38 

871  3  •• 

12868 

88707 

11293 

90268 

09732 

22 

39 

87158 

12842 

88733 

11267 

90294 

09706 

21 

40 

9.87185 

10.12815 

9.88759 

10.11241 

9.90320 

10.09680 

20 

41 

8721  1 

12789 

88786 

11214 

90346 

09654 

19 

42 

87238 

12762 

SSS12 

11188 

90371 

09629 

18 

43 

872(54 

12736 

88838 

11162 

90397 

09603 

17 

44 

87290 

12710 

88884 

11136 

90423 

09577 

16 

45 

87317 

12683 

88890 

11110 

90449 

09551 

15 

4(5 

87343 

12657 

88916 

11084 

90475 

09525 

14 

47 

87369 

12631 

88942 

11058 

90501 

09499 

13 

48 

87396 

12604 

88968 

11032 

90527 

09473 

49 

87422 

12578 

88994 

11006 

90553 

09447 

11 

50 

9'.  87448 

10.12552 

9.89020 

10.10980 

9.90578 

10.09422 

10 

51 

87475 

125  25 

8904(5 

10954 

90604 

09396 

9 

52 

87501 

12499 

89073 

10927 

90630 

09370 

H 

53 

87527 

12473 

89099 

10901 

90656 

09344 

7 

54 

87554 

12441! 

89125 

10875 

90682 

OSS  18 

6 

55 

87580 

12420 

89151 

10849 

90708 

09292 

5 

56 

87606 

12894 

89177 

10823 

90734 

09266 

4 

57 

87633 

12367 

89203 

10797 

90759 

09241 

3 

58 

87659 

12341 

89229 

10771 

90785 

09215 

2 

59 

87685 

12315 

89255 

10745 

90811 

09189 

1 

60 

87711 

12289 

89281 

10719 

90837 

09163 

0 

f 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

53° 

52° 

51° 

266   TABLE  VIII. -LOG.  TANGENTS    AND   COTANGENTS- 


/ 

39° 

40° 

41° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.90837 

10.09163 

9.92381 

10.07619 

9.93916 

10.06084 

60 

1 

90803 

09137 

92407 

07593 

93942 

06058 

59 

2 

90889 

09111 

92433 

07567 

93967 

06033 

58 

3 

90914 

09086 

92458 

07542 

93993 

06007 

57 

4 

90940 

09060 

92484 

07516 

94018 

05982 

56 

5 

90966 

09034 

92510 

07490 

94044 

05956 

55 

6 

90992 

09008 

92535 

07465 

94069 

05931 

54 

7 

91018 

08982 

92561 

07439 

94095 

05905 

53 

8 

91043 

08957 

92587 

07413 

94120 

05880 

52 

9 

91069 

08931 

92612 

07388 

94146 

05854 

51 

10 

9.91095 

10.08905 

9.92638 

10.07362 

9.94171 

10.058-29 

50 

11 

91121 

08879 

92663 

07337 

94197 

05803 

49 

12 

91147 

08853 

92689 

07311 

94222 

05778 

48 

13 

91172 

088-28 

92715 

07285 

91248 

05752 

47 

14 

91198 

08802 

92740 

07260 

94273 

05727 

46 

15 

9J224 

08776 

92766 

07234 

94299 

05701 

45 

16 

91250 

08750 

92792 

07208 

94324 

05676 

44 

17 

91276 

08724 

92817 

07183 

94350 

05650 

43 

18 

91301 

08699 

92843 

07157 

94375 

056-25 

42 

19 

91327 

08673 

92868 

07132 

94401 

05599 

41 

20 

9.91353 

10.08647 

9.92894 

10.07106 

9.94426 

10.05574 

40 

21 

91379 

08621 

92920 

07080 

94452 

05548 

39 

22 

91404 

08596 

92945 

07055 

94477 

055-23 

38 

23 

91430 

08570 

92971 

07029 

94503 

05497 

37 

24 

91456 

08544 

92996 

07004 

94528 

05472 

36 

25 

91482 

08518 

93022 

06978 

94554 

05446 

35 

26 

91507 

08493 

93048 

06952 

94579 

05421 

34 

27 

91533 

08467 

93073 

06927 

94604 

05396 

33 

28 

91559 

08441 

93099 

06901 

94630 

05370 

32 

29 

91585 

08415 

93124 

06876 

94655 

05345 

31 

30 

9.91610 

10.08390 

9.93150 

10.06850 

9.94681 

10.05319 

30 

31 

91636 

08364 

93175 

06825 

94706 

05294 

29 

32 

91662 

08338 

93201 

06799 

94732 

05268 

28 

33 

9168S 

08312 

93227 

06773 

94757 

05243 

27 

31 

91713 

08287 

93252 

06748 

94783 

05217 

26 

35 

91739 

08261 

93278 

06722 

94808 

05192 

25 

36 

91765 

08235 

93303 

06697 

94834 

05166 

24 

37 

91791 

08209 

93329 

06671 

94859 

05141 

23 

38 

91816 

08184 

93354 

06646 

94884 

05116 

22 

98 

91842 

08158 

93380 

06620 

94910 

05090 

21 

40 

9.91868 

10.08132 

9.93406 

10.03594 

9.94935 

10.05065 

20 

41 

91893 

08107 

93131 

06569 

94961 

05039 

19 

4-2 

91919 

08081 

93457 

06543 

94986 

05014 

18 

43 

91945 

08055 

93482 

06518 

95012 

04988 

17 

44 

91971 

OS029 

93508 

06492 

95037 

04963 

16 

45 

91996 

08004 

93533 

06467 

95062 

04938 

15 

46 

92022 

07978 

93559 

06441 

95088 

04912 

14 

47 

92048 

07952 

93584 

06416 

951  13 

04887 

13 

48 

92073 

07927 

9W10 

06390 

95139 

048(51 

12 

49 

92099 

07901 

93636 

0636  1 

95164 

04836 

11 

50 

9.92125 

10.07H75 

9.93661 

10.06'i39 

9.95190 

10.04810 

10 

51 

92150 

07B50 

93687 

06313 

95215 

04785 

9 

52 

92176 

07824 

83712 

06288 

95240 

04760 

8 

f>3 

92202 

077'98 

93738 

0626-2 

95266 

04734 

54 

92227 

07773 

93763 

06237 

95-291 

04709 

6 

55 

92253 

07747 

93789 

06211 

95317 

046J-8 

5 

56 

92279 

07721 

93S14 

06186 

9534-2 

04658 

4 

57 

9-2304 

07696 

93840 

06160 

95368 

04682 

3 

58 

92330 

07670 

93865 

06135 

95393 

04607 

2 

59 

92356 

07644 

93891 

06109 

95418 

04682 

1 

60 

92381 

07619 

93916 

06084 

95444 

04556 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

60° 

49° 

48° 

K  vin.— LOG.  TANGENTS  AND  COTANGENTS.  2 


4' 

jo              430 

44° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.95444 

10.04556 

9.96966 

10.03034 

9.98484 

10.01516 

60 

1 

95469 

04531 

90991 

03009 

98509 

01491 

59 

o 

95495 

04505 

97026 

02984 

98534 

01466 

58 

3 

95520 

04480 

97042 

02958 

98560 

01440 

57 

4 

95545 

04455 

97067 

02933 

98585 

01415 

56 

5 

95571 

04429 

97092 

02908 

98610 

01390 

55 

6 

95596 

04404 

97118 

02882 

98035 

01365 

54 

7 

95622 

04378 

97143 

02857 

98661 

01339 

53 

g 

95047 

04353 

97168 

02832 

98686 

01314 

52 

9 

95072 

04328 

97193 

02807 

98711 

01289 

51 

10 

9.95698 

10.04302 

9.97219 

10.02781 

9.98737 

10.01263 

50 

11 

95723 

04277 

97244 

02756 

98762 

01238 

49 

12 

95748 

04252 

97269 

02731 

98787 

01213 

48 

13 

95774 

04226 

97295 

02705 

98812 

01188 

47 

14 

95799 

04201 

97320 

02680 

98838 

01102 

46 

15 

95825 

04175 

97345 

02655 

98863 

0113: 

45 

16 

95850 

04150 

97371 

02629 

98888 

01112 

44 

17 

95875 

04125 

97396 

02604 

98913 

01087 

43 

18 

95901 

04099 

97421 

02579 

98939 

01001 

42 

19 

95926 

04074 

97447 

02553 

98964 

01036 

41 

20 

9.95952 

10.04048 

9.97472 

10.02528 

9.98989 

10.01011 

40 

21 

95977 

04023 

97497 

02503 

99015 

009S5 

39 

22 

96002 

03998 

97523 

02477 

99040 

00960 

38 

23 

96028 

08972 

97548 

02452 

99065 

00935 

37 

24 

90053 

03947 

97573 

02427 

99090 

00910 

36 

25 

96078 

03922 

97598 

02402 

99116 

00884 

35 

26 

96104 

03896 

97624 

02370 

99141 

00859 

34 

S3 

96129 

03871 

97649 

02351 

99166 

00834 

33 

28 

96155 

03845 

97674 

02326 

99191 

00809 

32 

29 

96180 

03820 

97700 

02300 

99217 

00783 

31 

30 

9.96205 

10.03795 

9.97725 

10.02275 

9.99242 

10.00758 

30 

31 

96231 

03769 

97750 

02250 

99267 

00733 

29 

32 

96256 

03744 

97776 

02224 

99293 

00707 

28 

33 

96281 

03719 

97801 

02199 

99318 

00682 

27 

34 

96307 

03693 

97826 

02174 

99343 

00657 

26 

35 

96332 

03668 

97851 

02149 

99368 

00632 

25 

36 

96357 

03643 

97877 

02123 

99394 

00606 

24 

37 

96383 

03617 

97902 

02098 

99419 

OC581 

23 

38 

96408 

03592 

97927 

02073 

99444 

00556 

22 

39 

96433 

03567 

97953 

02047 

99469 

00531 

21 

40 

9.96459 

10.03541 

9.97978 

10.02022 

9.99495 

10.00505 

20 

41 

96484 

03516 

98003 

01997 

99520 

00480 

19 

42 

96510 

03490 

98029 

01971 

99545 

00455 

18 

43 

96535 

03465 

98054 

01946 

99570 

00430 

17 

44 

96560 

03440 

98079 

01921 

99596 

00404 

16 

45 

90&6 

03414 

98104 

01896 

99621 

00379 

15 

46 

90011 

03389 

98130 

01870 

99646 

00354 

14 

47 

96630 

03364 

98155 

01845 

99672 

003:28 

13 

48 

96062 

03338 

98180 

01820 

99697 

00303 

12 

49 

96687 

03313 

98206 

01794 

99722 

00278 

11 

50 

9.96712 

10.03288 

9.98231 

10.01769 

9.99747 

10.00253 

10 

51 

96738 

03262 

98256 

01744 

99773 

002-J7 

9 

52 

96763 

03237 

98281 

01719 

99798 

00202 

8 

53 

96788 

03212 

98307 

01693 

99823 

00177 

7 

54 

96814 

03186 

98332 

01068 

99848 

00152 

6 

55 

96839 

03161 

98357 

01643 

99874 

00126 

5 

56 

96864 

03136 

98383 

01017 

99899 

00101 

4 

57 

90890 

03110 

98408 

01592 

99924 

00076 

3 

58 

90915 

03085 

98133 

01567 

99949 

00051 

2 

59 

96940 

03060 

98458 

01542 

99975 

00025 

1 

60 

96966 

03034 

98484 

01516 

10.00000 

00000 

0 

; 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

47° 

46° 

45° 

268    A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


TABLE  IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

The  Long  Chords,  Mid-Ordinates,  Externals,  and  Tangent 
Distances  of  this  table  are  for  a  curve  of  5780  feet  radius.  To 
find  the  corresponding  functions  of  any  other  curve  divide  the 
tabular  values  by  the  degree  of  curve. 

For  metric  curves  having  20-metre  chords,  multiply  tne  degree 
by  5  and  enter  the  table  with  the  result  as  a  value  of  D,  the  tabu- 
lar values  being  taken  as  metres  instead  of  feet 

Thus  for  a  1°  30'  metric  curve  having  1=  45  the  tangent  dis- 
tance is  T=  7-5-77-^  =  316.45  metres.  Again,  suppose  /=  38° 

and  the   long  chord  =  373.1  m.  known  and  D  required.     The 

3731.0 


tabular  L.  C.  is  3731  m. ;  therefore  D  = 


378: 1  X  5 


0° 

1° 

1 

L.  C.       M.            E.           T. 

L.  C.        M.           E.           T. 

0 

0.00     0.000      0.000        0.00 

100.00     0.218     0.218       50.00 

0 

2 

3.33     0  000      0.000        1  .07 

103.33     0.233     0.233       51  67 

2 

4 

6.6?     0001      0.001        3.33 

100.66     0.248     0.248       53.33 

4 

6 

10.00     0.002      0.002        5.00 

110.00     0.264     0.264       55.00 

6 

8 

13.33     0.004      0.004        6.67 

113.33     0.280     0.280       56.6? 

8 

10 

16.67     0  006      0.006        8.33 

116.66     0.297     0.29?       58.33 

10 

12 

20.00     0.009      0.009      10.00 

120  00     0.314     0.314       60  (X) 

12 

14 

23.33     0.012      0.012      11.67 

123.33     0.332     0.332       61.67 

14 

16 

26.67     0  015      0.015      13.33 

126.66     0.350     0.350       63  33 

16 

18 

30.00     0.019      0.019      15.00 

130.00     0.368     0.368       65.00 

18 

20 

33.33     0.024      0.024      10.07 

133.33     0.388     0.388       66  67 

20 

22 

3«  -07     0.029      0.029      18.33 

130.06     0.407     0.407        68.33 

22 

24 

40.00     0.035      0.035      20.00 

140.00     0.427     0.42?       70.00 

24 

26 

43.33     0.041      0.041      21.67 

143.33     0.448     0.448       71.6? 

26 

28 

46.67     0.048      0.048      23.33 

146.66     0.409     0.409       73.33 

28 

30 

50.00     0  054      0.054      25.00 

150.00     0.491      0  491        75.00 

30 

32 

53  33     0  062      0.002      26.67 

153.33     0.513     0.513       70.07 

32 

34 

56  67     0  070      0  070      28.33 

150.60     0.536     0  536       78.33 

34 

36 

60.00     0.079      0.07!)      30.00 

160.00     0.559     0.559       80.00 

36 

38 

63.33     0.088      0088      31.67 

163.33     0.582     0.582       81.67 

38 

40 

66.67     0  097      0.097      33  33 

166  66     0.606     0.606       83.33 

40 

42 

70.00     0.107      0.107      35.00 

170.00     0.630     0.630       85  00 

42 

44 

73.33     0.117      0.117      36.07 

173  33     0.655     0.655        86.67 

44 

46 

76.6?      0  128      0.128      38.33 

170.66     0.681      0.681        88.33 

46 

48 

80.00     0.140      0.140      40.00 

ISO.  00     0.706     0.706        90.00 

48 

50 

83.33     0.151       0.151       41.67 

1S3  33     0  733     0.733       91.67 

50 

52 

86  67     0.164      0.164      43..°,:! 

180.66      0.700      0.700        «.»3.:tt 

52 

54 

90  00     0  176      0.176      45  00 

190  IK)     0.7SS     0  7S8       1)5.  00 

54 

56 

93  33     0.190      0.190      46.67 

193.33     0.815     0.815       %  67 

56 

58 

96.67     0.204       0.204      48  33 

190  66     0.844     0.844       98.33 

58 

60 

100.00     0.218      0.218      50  00 

199.98     0.873     0  873     100.00 

60 

IX.— FUNCTIONS   OF    A   ONE-DEGREE   CURVE.     269 


2° 

3° 

f 

L.  C.         M.          E.           T. 

L.  C.          M.           E,            T. 

/ 

0 

199  98      0  873      0.873    100.00 

299.96      1.964      1.964     150.07 

0 

2 

203.31       0.902      0.902    101.67 

303.29      2.008      2.009     151.74 

2 

4 

206.64      0  932      0.932    103.34 

300.62      2.053      2.054     153.41 

4 

6 

2C9.97      0.962      0.962    105.01 

30995      2.098      2.099     155.08 

6 

8 

213.31       0.993      0.993    106.68 

313.29      2.143      2.144     156.75 

8 

10 

216.64      1.024      1.024    108.35 

316.62      2.188      2.189     158.42 

10 

12 

219.97       1.056      1.056    110.02 

319.95      2.235      2.236     160.09 

12 

14 

223.30      1.088      1.088    111.69 

323.28      2.282      2.283    161.76 

14 

16 

226.64       1.121      1.121     113.36 

326.62      2.329      2.330     163.43 

16 

18 

229.97      1.154      1.154    115.02 

329.95      2.376      2.377     165.09 

18 

20 

233.30         .188      1.188    116.69 

333.28      2.424      2.425    166.76 

20 

22 

236.63         .222       1.222     118.36 

336  61      2.473      2.474     168.43 

22 

24 

239  97         .256      1.256    120.03 

339.95      2.523      2.523     170.10 

24 

26 

243.30         .292      1.292    121.70 

343.28      2.572      2.573     171.77 

26 

28 

246.63         .328      1.328    123.37 

346.61      2.622      2.623     17344 

28 

30 

249.96         .364      1.364     125.03 

349.94      2.672      2.673    175.10 

30 

32 

253.29         .399      1.399    126.70 

353.27      2.724      2.725     176.72 

32 

34 

256  62         .437      1.437    128.37 

356.60      2.776      2.777     178.39 

34 

36 

259.96         .475      1.475    130.04 

359.94      2.828      2.829     18006 

36 

38 

263.29         .513      1.513    131.71 

363.27      2.880      2.881     181.73 

38 

40 

266.62         .552      1.552    13338 

366  60      2  933      2.934     183.40 

40 

42 

269.96         .592      1.592    135.05 

369.94      2.987      2.988     185.07 

42 

44 

273.29         .632      1.632     136.72 

373.27      3.042      3  043     186.74 

44 

46 

276  62          672      1.672     138.38 

376.60      3.096      3097     188.40 

46 

48 

279.96         .712      1.712    140.05 

379.94      3.151      3.152    190.07 

48 

50 

283.29         .752      1.752     141.72 

383.27      3.206      3.207     191.74 

50 

52 

2S6.62         .794      1.794     14339 

38660      3.263      3.264     193.41 

52 

54 

2S9  96         .836      1.836    145.06 

389.94      3.320      3.321     195.08 

54 

56 

293.29         .878      1.878     146.73 

393.27      3.377      3  378    196.75 

56 

58 

296.62         .921       1.921     148.40 

396.60      3.434       3.435     198.42 

58 

60 

299.96         .964      1.964     15007 

399.94      3.491       3.492    2uO  .09 

60 

4° 

5° 

/ 

L.  C.         M.           E.          T. 

L.  C.         M.          E.          T. 

0 

399  94      3.491       3.492    200.09 

499.88      5.454      5.459    250.17 

0 

2 

403.27      3.550      3.551     201.76 

503.21       5.^27      5.o33     251.84 

2 

4 

406.60      3.609      3.610    203.43 

506.54      5.601      5.607    253.51 

4 

6 

409.93      3.668      3.670    205.10 

509.87      5.675      5.681     255.18 

6 

8 

413.26      3.727      3.730    206.77 

513.20      5.749      5.755    256.85 

H 

10 

416  59      3.787      3.790    208.44 

516  53      5.823      5.829    258.52 

10 

12 

41992      3.848      3.851     210.11 

519.86      5.899      5.905    260.20 

12 

14 

423.26      3.910      3.913    211.77 

523.19      5.975      5981     261.86 

14 

16 

426.59      3.972      3.975    213.45 

526.52      6.052      6.058    263.54 

16 

18 

429.92      4.034      4.037    215.11 

529.85      6.129      6.135    265.20 

18 

20 

433.25      4.096      4.099    216.78 

533.18      6.206      6.212    266.87 

20 

22 

436.58      4.160      4.163    218.45 

536.51      6.284      6.290    268.54 

22 

24 

439.91       4.224      4.227    220.12 

539.84      6.362      6.369    270.21 

24 

26 

443.24      4.288      4.291     221.79 

543.17      6.441      6.448    271.88 

26 

28 

446.58      4.353      4.356    223.46 

546.50      6.520      6.527    273.54 

28 

80 

449.91       4.418      4.421     225.13 

549.83      6.599      6.606    275.21 

30 

32 

453.24      4.484      4.487    226.80 

553  17      6.680      6.687    276.88 

32 

34 

456.57      4.550      4.554    228.47 

556.50      6.761       6.768    278.55 

34 

315 

459.90      4.617      4.621     230.14 

559.83      G.842      6.849    280.23 

36 

38 

463.23      4.684      4.688    231.81 

563.16      6923      6.931     281.90 

38 

40 

466.56      4.751      4.755    233.48 

566.49         .005          013    283.57 

40 

42 

469.8!)      4.820      4.824    235.15 

569.82        .088         .096    285.24 

42 

44 

473  23      4.889      O..S93    236.82 

573.15          171         .180    286.91 

44 

46 

476  56      4.958      4  962    23S.48 

576.48          255         .264    288.59 

46 

48 

479.89      5.027      5.031     2-10.15 

579.81          339         .348    290.26 

48 

50 

483.22      5.096      5100    241  82 

583.14          423         .432    291-93 

50 

52 

486.55  •    5.167      5.171     243.49 

586.47          508         .517    293.60 

52 

54 

489  88      5.238      5.243    2-15.16 

nS'.KKO         .593         ,603    295.27 

54 

50 

493.21       5.310      5.315    2)6  83 

593.13          678          689    296  95 

56 

58 

49(154      5.382      5387     24850 

596.46         .764          775    298.62 

58 

60 

499.88      5  454      5.459     250.17 

599.80      7.850          861     300.30 

60 

270     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


6° 

4  ° 

/ 

L   C.          M.            E          T. 

L.  C.         M.           E.           T. 

/ 

0 

599.80      7.850      7.861    300.30 

Will.  00      10.09     10.71     350.44 

0 

2 

603.13      7.940      7.951     301.97 

702.93      10.79     10.81      3.r>2.11 

2 

4 

606  46      8.030      8  041    303.04 

700.26      10.90     10.92     353.79 

4 

6 

609.78      8.120      8.131     305.31 

70958     11.00     11.02     35540 

6 

8 

613.11       8.210      8.2v!l     3iH>  US 

712.91      11.11     11.13     357.13 

8 

10 

616  44      8.300      8.311    308.05 

716.24     11.21     11.23     358  81 

10 

12 

619.76      8.390      8.401     310.82 

71956     11.31     11.33     360.48 

12 

14 

623.09      8.480      8.491    311.99 

722.89     11.42     11.44     362.15 

14 

16 

626.42      8.570      8.581    313.66 

720  21      11.52     11.54     363  83 

16 

18 

629.74      8.660      8.671     315.33 

729.53     11.63     11.65     365.50 

18 

20 

633.07      8.750      8.761    317.00 

732.86      J1.73     11.75     367.17 

20 

22 

636.40      8.844      8.856    318.67 

736.19      11.84     11.86     368.85 

22 

24 

639.72      8.939      8.951     320.31 

739.51      11.95     11.97     370.52 

24 

26 

643.05      9.033      9.046    322.01 

74281      12.06     12.08     372.19 

26 

28 

646.38      9.128      9.141    323.08 

740.17     12.17     12.19     373.80 

28 

30 

649.70      9.222      9.236    325  35 

7'49.49     12.27     12.30     375.54 

30 

32 

653.03      9.317      9.331     327.02 

752.82     12.38     12.41     377.22 

32 

34 

656.36      9.411      9.426    328.09 

750.15      12.49     12.52     378.89 

34 

36 

659.69      9.506      9.521     330.37 

759.47     12.60     12.63     380  57 

36 

38 

663.02      9.600      9.616    332  04 

762.80     12.71      12.74     382.24 

38 

40 

666.34      9.095      9.712    333.71 

766.13     12.82     12.85     383.92 

40 

42 

669.67      9.794      9.812    335.38 

769.45     12.93     12.96     385.00 

42 

44 

673.00      9.894      9.912    337.05 

772.78     13.04     13.08     387.27 

44 

46 

676.32      9.993     10.01      338.73 

776.11      13.15     13.19     388.95 

46 

48 

679.65    10.09      10.11      340.40 

779.43     13.26     13.31     390.62 

48 

50 

682.98     10.19      10.21      342.07 

782.76     13.37     13.42     392.30 

50 

52 

686.30    10.29      10.31       343.74 

786.09     13.48     13.53     393.98 

52 

54 

689.63     10.39      10.41      345.41 

789.41      13.59     13.05     395.65 

54 

56 

692.96     1049      10.51      347.08 

792.74     1370     13.76     397.  33 

56 

58 

696.28     10.59      10.61      348.76 

796.07      13.81     13.88     399.01 

58 

60 

099  00    10.69      10.71      350.44 

799.40     13  96     13.99     400.70 

60 

8° 

9° 

L.  C.        M.           E.           T. 

L.  C.        M.          E.           T. 

0 

799.40     13.96       13.99     400.70 

899.10     17.06     17.71     450.95 

0 

2 

802.72     14.07       14.10     402.37 

902.42     17.79     17.84    452.63 

2 

4 

806.04     14.19       14.22     404.05 

905.74     17.92     17.98    454.31 

4 

6 

809.37     1431       14.34     405.72 

909.07     18.06     18.11    455.98 

6 

8 

812.69     14  43       14.46     407.39 

912.39     18.19     18.25    457.66 

8 

10 

81G.01      14.55       14.58     409.06 

915.71      18.32     18.38    459.34 

10 

12 

819.34     14.66       14.70     410.74 

919.04     18.46     18.52    461.02 

12 

14 

822.66     14.78       14.82     412.41 

922.36     18.59     18.65    462.70 

14 

16 

825.98     14.90       14.94     414.03 

925.68     18.72     18.79    404.37 

16 

18 

829.31      15.02       15.06     415.75 

929.01      18.86     18.92    466.05 

18 

20 

83-,'.  63     15.14      15.18     417.43 

932.33     18.99     19.06    467.73 

20 

22 

835.95     15.26      1530     419.10 

935.65     19.12     19.19    469.41 

22 

24 

839.28     15.38       15  43     4J0.77 

938.98     19.26     19.33    471.08 

24 

26 

842.60     15.51       15.55     422.45 

942.30     19.40     19.47    472.70 

26 

28 

845.92     15.63       15.08     424.12 

945.62     19.54     19.01     474.43 

28 

30 

849.25     15.75       15.80     425.79 

948.95     19.68     19.75    476.10 

30 

32 

852.57     15.88       15.93     427.47 

952.27     19.82     19.89    477.78 

32 

34 

855.89     16.00      16.05     429.15 

955.59     19.96     20.03    479.46 

34 

36 

859.22     16.12       16.18     430.82 

958.92     20.10     20.17    481.14 

36 

38 

862.54     16.25       16.30     432.50 

962.24     20.24     20.31    482.83 

38 

40 

865.86     16.38       16.43     434.18 

965.56     20.38     20.45    484.51 

40 

42 

869.19     16.50      16.55     435.86 

908.89     20.52     20.59    480.19 

42 

44 

872.51      16.63       16.68     437.54 

972.21      20.66     20.74    487.87 

44 

46 

875.83     16.76       16.81      439.21 

975.53     20.80     20.88    489.56 

46 

48 

879.16     16.89       16.94     440.89 

5»7«  80     20.94     21.03    491.24 

48 

50 

882.48     17.02           .07     442.57 

982.18     21.09     21.17    492.92 

50 

52 

885.80     17.14           .19     444.25 

985.50     21.23     21.31    494.60 

52 

54 

889.13     17.27           .32     445.93 

988.83     21.37     21.46    496.28 

54 

56 

892.45     17.40           .45     447.60 

992.15     21.51     21.60    497.96 

56 

58 

895.77     17.53           .58     419.28 

995.47     21.65     21.75    499.65 

58 

60 

899.10     17.66           .71      450.95 

998.80     21.80     21.89    501.32 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     27i 


10° 

11° 

L  C.         M.           E.          T. 

L.  C.         M.           E.          T. 

0 

998.8      21.80      21.89    501.32 

1098.4     26.38      26.50     551  74 

0 

1002.1      21.94      22.03    503.00 

1101.7     26.54      26.66     553.42 

0 

4 

1005.4      22.09      22.18    504.68 

1105.0     26  70      20.83     555.10 

4 

6 

1008.8      22.24      22.33    500.30 

1108  3     26.86      26.99     556.78 

6 

8 

1012.1      22.39      22.48    508.04 

1111.7     27.02      27.16     558.46 

8 

10 

1015.4      22.54      22.63    509.12 

1115.0     27.19      27.32     560.14 

10 

tat 

1018.7      22.68      22.78    511.40 

1118.3     27.35      27.48     561.82 

12 

14 

1022  0      22.83      22.93    513.08 

1121.6     27.51      .27.65     563.50 

14 

16 

1025.4      22.98      23.08    514.76 

1124.9     27.67      27.81     565.18 

16 

18 

1028.7      23.13      23.23    516.44 

1128.2     27.83      27.98     566.86 

18 

20 

1032.0      23.28      23.38    518.12 

1131.6     28.00      28.14     568.54 

20 

Oj> 

1035.3      23.43      23.53    519.80 

1134.9     26.17      28.30     570.22 

22 

24 

1038.6      23.58      23  68    521.48 

1138.2     28.34      28.47     571.90 

24 

26 

1042.0      23.73      23.84    523.16 

1141.5     28.50      2864     573.58 

26 

28 

1045.3      23.88      23.99    524.85 

1144.8     28.67      28.81     575  27 

28 

30 

1048.6      24.04      24.14    526.53 

1148.1     28.84      28.98     570.95 

30 

32 

1051.9      24.19      24.30    528.21 

1151.5     29.00      29.14     578.63 

32 

34 

1055.2      24.34      24.45    529.89 

1154.8     29.17      29.31      580.32 

34 

36 

1058.6      24.49      24.60    531.57 

1158.1      29.34      29.48     582.00 

36 

38 

1061.9      24.64      24.76    533.25 

1161.4     29.50      29.65     583.69 

38 

40 

1065.2      24.80      24.91     534.93 

1164.7     29.67      29.82     585.37 

40 

42 

1008.5      24.95      25.06    536.61 

1168.0     29.84      29.99     587.05 

42 

44 

1071.8      25.11      25.22    538.29 

1171.4     30.01      30.17     588.74 

44 

46 

1075.2      25.27      25.38    539.97 

1174.7     30.18      30.34     590.42 

46 

48 

1078.5      25.43      25.54    541.65 

1178.0     30.35      30.52     592.11 

48 

50 

1081.8      25.59      25.70    543.33 

1181.3     30.53      30.69     593.79 

50 

52 

1085.1      25.74      25.86    545.01 

1184.6     30.70      30.86     595.47 

52 

54 

1088.4      25.90      26.02    546.69 

1187.9     30.87      31.04     597.16 

54 

56 

1091.8      26.06      26.18    548.37 

1191.3     31.04      31.21     598.84 

56 

58 

1095.1      26.22      20.34    550.00 

1194.6     31.21      31.39     600.53 

58 

60       1098.4      26.38      26.50    551.74 

1197.9     31.39      31.56     602.22 

60 

12° 

13° 

L.  C.         M.           E.          T. 

L.  C.         M.           E           T. 

0 

1197.9      31.39      31.50    002.22 

1297.3     36.83      37.07     652.87 

0 

2 

1201.2      31.57      31.13    603.91 

1300.6     37.02      37.26     654.56 

2 

4 

1204.5      31.74      31.91     605.00 

1303.9     37.21      37.46     656.25 

4 

6 

1207.8      31.92      32.09    607.28 

1307.2     37.40      37.65     657.93 

6 

8 

1211.1       32.09      32.27    608.97 

1310.5     37.59      37.85     659.62 

8 

10 

1214.5      32.27      32.45     610.66 

1313.8     37  79      38.04     661.31 

10 

12 

1217.8      32.45      32.63    612.35 

1317.2     37.98      38.23     663.00 

12 

14 

1221.1      32.62      32.81     614.04 

1320.5     38.17      38.43     664.69 

14 

16 

1224.4      32.80      32.99    615.72 

1323.8     38.36      38.62     666.37 

16 

18 

1227.7      32.97      33.17    617.41 

1327.1      38.55      38.82     668.06 

18 

20 

1231.0      33.15      33.35    619  10 

1330.4     38.75      39.01      669.75 

20 

22 

1231.3      33.33      33.53    620.79 

1333.7     38.95      39.20     671.44 

22 

24 

1237.7      33.51      33.72    022.48 

1337.0     39.15      39.40     613.13 

24 

26 

1241.0      33.69      33.  90    624  16 

1340.3     39.35      39.60     674.81 

26 

28 

1244.3      33.87      34.09    G-,'5.85 

1343.6     39.54      39.80     676.51 

28 

30 

1247.6      34.06      34.27     627  55 

1346.9     39.74      40.00     678.20 

30 

32 

1250.9      34.24      34.45    629.24 

1350.3     39  94      40.19     679.89 

32 

34 

1254.2      34.42      34  64    630  93 

1353.6     40.13      40.39     681.58 

34 

36 

1257.5      34.60      34.82    632.61 

1356.9     40.33      40.59     683.26 

36 

38 

1260.8      34.78      35.01     634.30 

1360.2     40.52      40.79     684.95 

38 

40 

1264.2      3497      35.19    635.99 

1363.5     40.71      40.99     686.64 

40 

42 

1207.5      35.16      35.37    637  68 

1366.8     40.91      41.19     688.33 

42 

44 

1270.8      35.34      35.56    639.37 

1370.1      41.11       41.40     690.02 

44 

46 

1274.1      35.53      35.75    641.05 

1373.4     41.31      41.60     691.70 

46 

48 

1277.4      35.71      35.94    642.74 

1376.7     41.51       41.81      693.39 

48 

50 

1280.7      35.90      36.13    644.43 

1380.0     41.71      42.01      695.08 

50 

52 

1284.0      30.09      36.31     646.12 

1383.4     41.91       4221      696.77 

52 

54 

12S7.4      36.27      3(i.50    647.81 

1380.7     42.11       42.42     698.40 

54 

56 

1290.7      36.46      30.09    649.49 

1390.0     42.31       42.02     700.14 

56 

58 

1894.0      36.64      36.88    651.18 

1393.3     42.51      42.83     701.83 

58 

60 

1297.3      36.83      ST.  07     652  87 

139G.6     42.71       43.03     703.53 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE   CURVE. 


14° 

lo° 

L.  C.        M.         E.          T. 

L.  C.        M.         E.           T. 

/ 

0 

13%.  6    42.71    43.03    703.53 

1495.9    49.02    49.44    754.35 

0 

2 

1399.9    42  92    4323    705.23 

1499.2    49.24    49.66     750.05 

•> 

4 

1403.2    43.12    43.44    706.92 

1502.5    49.40    49.89    757.74 

4 

6 

1406.5    43.33    43.65    708.62 

1505.8    49.08    50.11     759.44 

6 

8 

1409.8    43.53    43.86    710.31 

1509.1     49.90    50.34    761.13 

8 

10 

1413.1     43.74    44.07    712.01 

1512.4    50.12    50.56    762.83 

10 

12 

1416.5    43.94    44.28    713.71 

1515.7    50.34    50.78    764.53 

12 

14 

1419.8    44.15    44.49    715.40 

1519.0    50.56    51.01     766.22 

14 

16 

1423.1     44.35    44.70    717.10 

1522.3    5078    51.23    767.92 

16 

18 

1426.4    44.56    44.91     718.79 

1525.6    51.00    51.46    769.61 

18 

20 

1429.7    44.77  '45.12    720.49 

1528.9    51.22    51.68    771.31 

20 

23 

1433.0    44  98    45.33    722.20 

1532.2    51.44    51.90    773.01 

22 

24 

1436.3    45.19    45.54    723.89 

1535.5    51.67    52.13    774.70 

24 

26 

1439.  G    45.40    45.76    725.59 

1538.8    51.89    52.36    776.40 

26 

28 

1442.9    45.61     45.97    727.28 

1542.1     52.12    52.59    778.09 

28 

30 

1446.2    45.82    40.18    728.97 

1545.4    52.34    52.82    779.79 

30 

32 

1449.6    46.03    46.40    730.66 

1548.7    52.57    53.05    781.49 

32 

34 

1452.9    46.24    46.61     732.35 

1552.0    52.79    53.28    783.19 

34 

86 

1456.2    46.45    46.82    734.05 

1555.3    53.02    53.51     784.89 

30 

38 

1439.5    46.66    47.04    735.74 

1058.6    53.24    53.74    786.59 

38 

40 

1462.8    46.87    47.25    737.43 

1561.9    53.47    53.97    788.29 

40 

42 

1460.  1     47.08    47.46    739.12 

1505.2    53.69    54.20    789.99 

42 

44 

1469.4    47.30    47.68    740.81 

1568.5    53.92    54.44    791.69 

44 

46 

1472.7    47.51     47.90    742.51 

1571.8    54.15    54.67    793.39 

46 

48 

1476.0    47.73    48.12    744.20 

1575.1     54.38    54.91     795.09 

48 

50 

1479.3    47.94    48.34    745.89 

1578.4    54.61     55.14    796.79 

50 

52 

1482.7    48.16    48.56    747.58 

1581.7    54.84    55.37    798.49 

52 

54 

1486.0    48.37    48.78    749.27 

1585.0    55.07    55.61     800.19 

54 

56 

1489.3    48.59    49.00    750.97 

1588.3    55.30    55.84    801.89 

56 

58 

1492.6    48.80    49.22    752.66 

1591.6    55.53    56.08    803.59 

58 

60 

1495.9    49.02    49.44    754.35 

1594.9    55.76    56.31     805.29 

60 

16° 

17° 

L.  C.        M.         E.          T. 

L.  C.        M.          E.           T. 

/ 

0 

1594.9    55.76    56.31     805.29 

1693.9    62.94    63.64    856.35 

0 

2 

1598.2    55  99    56.54     800.99 

1097.2    63.18    63.89    858.05 

2 

4 

1601.5    56.^3    56.78    808.64 

1700.5    63.43    64.15    859.76 

4 

6 

1604.8    56.46    57.02    810.39 

1703.8    63.68    64.40    861.46 

6 

8 

1608.1     56.70    57.26    812.09 

1707.1     63.93    64.06    863.16 

8 

10 

1611.4    56.93    57.50    813.79 

710.4    64.18    64.91     864.87 

10 

12 

1614.7    57.17    57.74    815.49 

713.7    64.42    65.16    866.57 

12 

14 

1618.0    57.40    57.98    817.19 

710.9    64.67    65.42    868.27 

14 

16 

1621.3    57.64    58.22    818.89 

720.2    64.92    65.67    869.98 

16 

18 

1624.6    57.87    58.46    820.59 

723.5    65.17    65.93    871.68 

18 

20 

1627.9    58.11     58.70    822.29 

726.8    65.42    66.18    873.38 

20 

22 

1631.2    58.34    58.94    823.99 

730.1     65.67    66.43    875.09 

22 

24 

1634.5    58.58    59  19    825.69 

733.4    65.93    66.69    876.79 

24 

26 

1637.8    58.82    59.43    827  39 

736.7    66.18    60.95    878.49 

26 

28 

1641.1     59.06    59.68    829.09 

740.0    66.44    67.21     880.20 

28 

80 

1644.4    59.30    59  92    830  79 

743.3    66.69    67.47    881.90 

30 

32 

1647.7    59.54    60.16    832.49 

746.0    60.94    67.72    883.61 

32 

34 

1651.0    59.78    6041     834.20 

749.9    67.20    67.98    885.32 

34 

36 

1654.3    60.02    60.65    835.90 

753.2    67.45    68.24    H87.02 

36 

38 

1657.6    60.26    60.90    837.61 

756.5    67.71     68.50    888.73 

38 

40 

16609    60.50    61.14    839.31 

759.8    67.93    68.76    890.44 

40 

42 

1664.2    60.74    61.39    841.01 

763.1     68.21     69.03    892.15 

42 

44 

1667.5    60.99    61.64    842.72 

766.3    68.47    69.29    893.86 

44 

46 

1670.8    61.23    61.89    844.42 

769.6    68.73    69.56    895.50 

46 

48 

1674.1     61.48    62.14    846.13 

772.9    68.99    69.82    897.27 

48 

50 

1677.4    61.72    62.39    847.83 

770  2    69.25    70.09    8'.)S.'.)S 

50 

52 

1680.7     61.96     6-J.04     849.53 

1779.5    69.50    70.36    900.69 

52 

54 

1684.0    62.21     62.89    851  24 

1782  8    69.70    70.02    902.40 

54 

56 

1087.3    62.45    63.14    852.01 

1780.1     70.02     70.89     904.10 

56 

58 

1690  6    62.70     03  39     S54.65 

17S9.4     70.  28     51.15     905.81 

58 

CO 

1693.9    62.  9  »     03  fi4     856  35 

1792.7     70.54     71.42     90^.."-2 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE    CURVE.     273 


18° 

19° 

' 

L.  C.        M.          E.            T. 

L.  C.       M.          E.           T. 

/ 

0 

1792.7    70.54    71.42      907.52 

1891.5    78.58    79.65      958.86 

0 

2 

1796.0    70.80    71.69      909.23 

1894.8    78.86    79.94      960.57 

2 

4 

1799.3    71.06    71.96      910.94 

1898.1     79.13    80.22      962.30 

4 

6 

1802.6    71.33    72.23      912.65 

1901.3    79.41     80.51       964.00 

6 

8 

1805.9    71.59    72.50      914.36 

1904.6    79.68    80.79      965.72 

8 

10 

1809.2    71.85    72.77      916.07 

1907.9    79.96    81.08      967.43 

10 

12 

1812.5    72.12    73.04      917.78 

1911.2    80.24    81.37      969.15 

12 

14 

1815.7    72.38    73.31      919.49 

1914.5    80.51     81.65      970.86 

14 

16 

1819.0    72.64    73.58      921.20 

1917.8    80.79    81.94      972.58 

16 

18 

1822.3    72  91     73.85      922.91 

1921.0    81.07    82.22      974.29 

18 

20 

1825.6    73.17    74.12      924.03 

1924.3    81.35  .82.51       976.01 

20 

22 

1828.9    73.43    74.39      926.34 

1927.6    81.63    82.80      977.72 

22 

24 

1833.  a    73.70    74.67      928.05 

1930.9    81.91     83.09      979.44 

24 

26 

1835.5    73.97    74.94      929.76 

1934.2    82.20    83  38      981.15 

26 

28 

1838.8    74.24    75.22      931.47 

1937.5    82.48    83.67      982.86 

28 

30 

1842.1     74.51     75.49      933.18 

1940.7    82.76    83.97      984.58 

30 

32 

1845.4    74.77    75.77      934.  ^9 

1944.0    83.05    84.26      986.30 

32 

34 

1S4S.7    75.04    76.04      936.60 

1947.3    83.33    84.55      988.02 

34 

36 

1852.0    75.31     76.32      938.32 

1950.6    83.61     84.84      989.74 

36 

38 

1855.3    75.58    76.59      940.03 

1953.9    83.90    85.13      991.46 

38 

40 

1858.6    75.85    76.87      941.74 

1957.2    84.18    85.43      993.18 

40 

42 

1861.9    76.12    77.14      943.45 

1960.4    84.47    85.73      994.90 

42 

44 

1865.1     76.39    77.42      945.16 

1963.7    84.75    86.02      996.62 

44 

46 

1868.4    76.67    77.70      946.88 

1967.0    85.04    86.32      998.34 

46 

48 

1871.7    76.94    77.98      948.59 

1970.3    85.32    86.61     1000.0 

48 

50 

1875.0    77.21     78.26      950.30 

1973.6    85.61     86.91     1001.8 

50 

52 

1878.3    77.49    78.53      952.01 

1976.9    85.90    87.21     1003.5 

52 

54 

1881.6    77.76    78.81      953.72 

1980.1    86.19    87.50    1005.2 

54 

5(5 

1884.9    78.03    79.09      955.44 

1983.4    86.47    87.80    1006.9 

56 

58 

1888.2    78.31     79.37      957.15 

19S6.7    86.76    88.09    1008.6 

58 

60 

1891.5    78.58    79.05      958  86 

1990.0    87.05    88.39    1010.4 

60 

20° 

21° 

L.  C.       M.          E.           T. 

L.  C.        M.         E.           T. 

/ 

0 

1990.0    87.05    88.39      1010.4 

2088.5      95.95      97.58    1062.0 

0 

2 

1993.3    87.34    88.69      1012.1 

2091.8      96.26      97.90    1063.7 

2 

4 

1996.6    87.63    88.99      1013.8 

2095.0      96.56      98.21     1065.4 

4 

6 

1999.8    87.92    89.29      1015.5 

2098.3      96.87      98.53    1067.2 

6 

8 

2003.1     88.21     89.59      1017.2 

2101.6      97.17      98.84     1068.9 

8 

10 

2006.4    88.50    89.89      1019.0 

2104.9      97.48      99.16    1070.6 

10 

12 

2009.7    88.79    90.19      1020.7 

2108.1      97.79      99.48    1072.4 

12 

14 

2013.0    89.08    90.49      1022.4 

2111.4      98.09      99.79    1074.1 

14 

16 

2016  3    89.37    90.79      1024.1 

2114.7      98.40     100.1      1075.8 

16 

18 

2019.5    89.66    91.09      1025.8 

2118.0      98.70     100.4      1077.5 

18 

20 

2022.8    89.96    91.40      1027.6 

2121.2      99.00     100.7      1079.3 

20 

22 

2026.1     90.25    91.71       1029.3 

2124.5      99.30     J01.1       1081.0 

22 

24 

2029-4    90.55    92.01       1031.0 

2127.8      99.60     101.4      1082.7 

24 

26 

2032.7    90.85    92.32      1032.7 

2131.0      99.90     101.7      1084.4 

26 

28 

2036.0    91.15    92.62      1034.4 

2134.3    100.2      102.0      1086.2 

28 

30 

2039.2    91.45    92.93      1036  1 

2137.6    100.5      102.3      1087.9 

30 

32 

2042.5    91.74    93.24      1037.9 

2140.9    100.8      102.7      1089.6 

32 

34 

2045.8    92.04    93.54      1039.6 

2144.1     101.1      103.0      1091.3 

34 

36 

2049.1    92.34    93.85      1041.3 

2147.4    101.4      103.3      1093.1 

36 

38 

2052  4    92.64    94.15      1043.0 

2150.7    101.7      103.6      1094.8 

38 

40 

2055.7    92.94    94.46      1044.8 

2154.0    102.1       104.0      1096.5 

40 

42 

2058.9    93.24    94.78      1046.5 

2157.2    102.4      104.3      1098.3 

42 

44 

2062.2    93.54    95.09      1048.2 

2160.5    102.7      104.6      1100.0 

44 

46 

2065.5    93.84    95.40      1049.9 

2163.8    103.0      104.9      1101.7 

46 

48 

2068.8    94.14    95.71       1051.7 

2167.1     103.3      105.3      1103.4 

48 

50 

2072.1     94.44    96.03       1053.4 

2170.3    103.6      105.6      1105.2 

50 

52 

2075.4     94.74     96.34      10T>5.1 

2173.6    103.9      105.9      1106.9 

52 

54 

2078.6    95.04    96.65      1056.8 

2176.9     104.2      106.3      1108.6 

54 

56 

2081.9     95.31     9696      1058.6 

2180.1     104.5      106.6      1110.3 

56 

58 

2085.2    95.64    97  27      1060.8 

2183.4    104.8      106.9      1112.1 

58 

60 

2088.5    95.95     97.58       1  062.0 

21867    105.2      107.2      1118.8 

60 

274      IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


22° 

2«° 

L.  C.        M.          E.          T. 

L.  0.        M.          E.         T. 

0 

218G.7     105.2     107.2    1113.8 

2284.8     115.0    117.4    1165.8 

0 

2 

2190.0     105.6     107.6    1115.5 

2288.1     115.3    117.7    1167.5 

2 

4 

:2193.2     105.9    107.9     1117.3 

2291.3    115.7     118.1     1109.2 

4 

6 

2196.5     106.2     108.2    1119.0 

2294.6     11(5.0    118.4    1171.0 

6 

8 

2199.8    106.5     108.6     1120.7 

2297.8    116.4     118.8    1172  7 

8 

10 

2203.0    106.8    108.9    1122.4 

2301.1     116.7    119.1     1174.4 

10 

12 

2206.3     107.1     109.2     1124.2 

2304.4    117.0    119.5    1176.2 

12 

14 

2209.6     107.4     109.6    1125.9 

2307.6    117.4     119.8    1177.9 

14 

16 

22129     107.7     109.9    1127.6 

2310.9    117.7    120.2    1179.7 

16 

18 

2216.1     108.0    110.2     1129.4 

2314.1     118.1     120.5    1181.4 

18 

20 

2219.4    108.4    110.6    1131.1 

2317.4    118.4    120.9    1183.1 

20 

22 

2222.7     108.7    110.9    1132.8 

2320.7     118.7    121.2    1184.9 

22 

24 

2225.9     109.0    111.2     1134.6 

23-23.9    119.1     121.6    1186.6 

24 

26 

2229.2    109.4    111.6     1136.3 

2327.2    119.4     121.9    1188.4 

26 

28 

2232.5     109.7    111.9    1138.0 

2330.4     119.8    122.3    1190  1 

28 

30 

2235.7     110.0    112.3    1139.7 

2333.7    120.1     122.6    1191.8 

30 

32 

2239.0    110.4     112.6     1141.5 

2337.0    120.4    123.0    1193.6 

i2 

34 

2242.3    110.7    112.9     1143.2 

2340.2    120.8    123.3    1195.3 

34 

36 

2245.6     111.0    113.3     1144.9 

2343.5     121.1     123.7    1197.1 

36 

38 

2248.8    111.4    113.6    1146.7 

2346.7    121.5    124.1     1198.8 

38 

40 

2252.1     111.7    113.9     1148.4 

2350.0    121.8    124.4    1200.5 

40 

42 

2255.4     112.0    114.3     1150.1 

2353.3     122.1     124.8    1202.3 

42 

44 

2258.6     112.3    114.6    1151.9 

2356.5    122.5    125.1    1204.0 

44 

46 

2261.9    112.7    115.0     1153.6 

2359.8    122.8     125.5    1205.8 

46 

48 

2265.2     113.0    115.3     1155.4 

23C.3.0    123.2    125.8    1207.5 

48 

50 

2268.4     113.3     115.7     1157.1 

23G6.3    123.5     126.2    1209  2 

50 

52 

2271.7    113.7    116.0    1158.8 

2369.6    123.8     126.6    1211.0 

52 

54 

2275.0     114.0     116.3     1160.6 

2372.8    124.2    126.9    1212.7 

54 

56 

2278.3    114.3     116.7     1162.3 

2376.1     124.5     127.3    1214.5 

56 

58 

2281.5     114.7     117.0     1164.0 

2379.3    124.9    127.6    1216.2 

58 

60 

2284.8     115.0     117.4     1165.8 

2382.6     125.2     1:28.0    1218.0 

60 

24° 

25° 

L.  O.        M.          E.         T. 

L  C.        M.          E.         T. 

0 

23S2.6    125.2    128.0    1218.0 

2480.4    135.8     139.1     1270.3 

0 

2 

2385.9    125.5    12S.4     1219.7 

2483.6    136.2     139.5     1272.0 

2 

4 

2389.1     125.9     128.7    1221.4 

2486.9    136.5    139.9    1273.8 

4 

6 

2392.4    126.2     129.1     12',>3.2 

24!!0.1     1369     140.3    1275.5 

6 

8 

2395.6    126.6     129.5    1224.9 

2493.4    137.2     140.6    1277.3 

8 

10 

2398.9    126.9    129.8     1226.7 

2496.6    137.6     141.0    1279.0 

10 

12 

2402.2    127.3     130.2    1228.4 

2499.9     1380    141.4     1280.8 

12 

14 

2405.4    127.6     130.6     1230.;> 

2503.1     138.3     141  8     1282.5 

14 

16 

2408.7    128.0    130.9    1231.9 

2506.4     138.7     142.2    1284.3 

16 

18 

2411.9    128.3    131.3     1233.6 

2509.6    139.0     142.5     1286.1 

18 

20 

2415.2    128.7     131.7    12354 

2512.9    139.4     142.9    1287.8 

20 

22 

2418.5    129.0    132.0    1237.1 

2516.1     139.8    143.3    1289.6 

22 

24 

2421.7     129.4    132.4    123-!.  9 

2519.4     140.1     143.7     1291.3 

24 

26 

2425.0    129.7    132.8    1240.6 

25:22.6    140.5     144.1     1293.1 

26 

28 

2428.2    130.1     133.1     1242.4 

2525.9    140.8     144.5    1294.8 

2S 

30 

2431.5     130.4    133.5    1244.1 

2529  1     141.2    114.9    12%.  0 

30 

32 

24:34.8     130.8     133.9     1215.8 

2532.4    141.6     145.3     1298.3 

32 

34 

2438.0    131.1     134.2    1247.6 

2535.6    1420    145.6    1300.1 

34 

36 

2441.3    131.5     134.6    1249.3 

2538.9    142.3     116.0    1301.8 

36 

38 

2444.5     131.8    135.0    1251.1 

2542.1     142  7    146.4     1303.6 

38 

40 

2447.8    132.2    135.4    1252.8 

2545.4    143.1     146  8    1305.3 

40 

42 

2451.1     132.6     135.7    1254.6 

2548.6    143  5     147.2    1307.1 

42 

44 

2454.3    132.9    136.1     1256.3 

2551.9    143.8     147.6     1308.8 

44 

46 

2457.6    133.3    136.5    1258.1 

2555.1     144.2    148.0     1310.6 

46 

48 

2460.  S    133.6    136.9    1259.8 

2558.4    144.5     148  4    1312.4 

48 

50 

2464.1     134.0    137.2    1261.5 

2561.6    144.9     148.8    1314.1 

50 

52 

2467.4     134.4    137.6    1263.3 

25(;i.9    145.3     149.2    1315.9 

52 

54 

2470.6     134.7    138.0    1265.0 

25(iS.l     145.7    149.5     1317.6 

54 

56 

2473.9     135.1     138.4    12668 

2571.4    140.0    14f».9     1319.4 

56 

58 

2477.1     135.4    138.7    12<»s.;i 

2574.6    146.4     150.3    1321.1 

58 

60 

2480.4     1358     139.1     1270.3 

2577.9     146  8     150.7     1322.  9 

r,o 

IX— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.       275 


26° 

27* 

L.  C.       M.         E.          T. 

L.  C.       M.         E.           T. 

/ 

0 

2577.9    146.8    150.7     1322.9 

2675  3     158.3     162.8     1375.6 

0 

2 

2581.1     147.1     151.1     1324.6 

2678.5     158.6    163.2    1377.4 

2 

4 

2584.4     147.5    151.5     1326.4 

2681.8    1590    163.7     1379.2 

4 

6 

2587.6     147.9    151.9     13281 

2685  0     159.4     164.1     1380.9 

6 

8 

2590.9     148.3    152.3     13^9.9 

2688.2     159.8     164.5    1882.7 

8 

10 

2594.1     148.7     152.7    1331.6 

2691.5     160.2    164.9    1384.5 

10 

12 

2597.4     149.1     153.1     1333.4 

2694.7     160.6    165.3    1386.2 

12 

14 

2600.6     149.4      53.5    1335.2 

2698.0    181  0    165.7    1388.0 

14 

16 

2603.9     149.8     153.9    1336.9 

2701.2     161.4     166  1     1389.8 

16 

18 

2607.1     1502    154.3    1338.7 

2704.4     161.8     1665    1391.5 

18 

20 

2610.4     1506    154.7    1340.4 

2707.7     162.2    167.0    1393.3 

20 

22 

2613.6     151  0     155.1     1342.2 

2710.9    162.6     167.4    1395.0 

22 

24 

26169    151.4     155.5     1.343.9 

2714.1     1630    167.8    1396.8 

24 

26 

2620.1       51.7     155.9    1345.7 

2717.4     163.4     168.2     1398.6 

26 

28 

2623.4     152.1     156.3    1347.4 

2720  6     163.8     168.6     1400.3 

28 

30 

2626.6     152.5    156.7    1349.2 

2723.8     164.2     169.1     1402.1 

30 

32 

2629.8    152.9    157.1     1351.0 

2727.1     164.6    169.5     1403.9 

32 

34 

2633.1     153.3     157.5    1352.7 

2730.3     165.0    169.9    1405.6 

34 

36 

2636  3     153.7     157.9    1354  5 

27336     165.4    170.3    1407.4 

36 

38 

2639.6    154.0    158.3    1356.2 

2736.8    165.8    170.8    1409.2 

38 

40 

2642.8     154  4     158.7    1358  0 

2740.0    166.2      71.2    1410.9 

40 

42 

2646.1     1.54.8     159.1     1359.8 

2743  3     166.6      71.6    1412.7 

42 

44 

2649.3     155.2     159.5    1361.5 

2746.5     167.0      72.0    1414.5 

44 

46 

2652.6    155  6    160.0    1363.3 

2749.7     167  4      72.5    1416.3 

46 

48 

2655.8    156.0    160.4     1365.1 

2753.0    167.8      72.9    1418.0 

48 

50 

2659.1     156.3    160.8    1366.8 

2756.2     168.2      73  3     1419.8 

50 

52 

2662.3     156.7    161.2    1368.6 

2759.5     168.6      73.7    1421.6 

52 

54 

2665.  C    157.1     161.6     13"0.4 

2762  7     169.0      74.1     1423.3 

54 

56 

2668.8    157  5     162  .0     13'  2.1 

27659     169.4      74.6    1425.1 

56 

58 

2672.1     157.9    162.4     1373  9 

2769.2     169.8      75.0    1426.9 

58 

60 

2675.3    158.3     162.8    1375.6 

2772.4     170.2      754    1428.6 

60 

SQfp 

29° 

L.  C.        M.         E           T. 

L.  C.        M.         E.          T. 

0 

2772.4     170.2    175  4    1428.6 

2869.4    182.5    188.5    1481.9 

0 

2 

2775.6     170.6      75.8     1430  4 

2872.6    182.9    189.0    1483.7 

2 

4 

2778.9     171.0      76.3     1432.2 

2875.8    183  3    189.4     1485.4 

4 

6 

2782.1       71.4       767     1434.0 

2879.1     183.7    189.9    1487.2 

6 

8 

27  K..  3      71.8      77.1     1435.7 

2*S2.3    184.2     190.3     1489.0 

8 

10 

27*8.6       72.2      77.6     1437.5 

2885.5    184.6    190.8    1490.8 

10 

12 

2791.8      72.6     178.0    1439.3 

2888  7     1S5.0     191.2     1492.6 

12 

14 

2795.0      730    178.4    1441.1 

2S92.0    185.4     191.7    1494.3 

14 

16 

2798.3      73.4     178.9    1442.8 

2S'.)5.2     185.8     192.1     1496.1 

16 

18 

2801.5       73.  S    179.3    1444.6 

2898.4    186.3    192.5    1497.9 

18 

20 

2804.7      74.3    179.7    1446.4 

2901.6     186.7     193.0    1499.7 

20 

"2 

2808.0      74.7    180.2    1448.2 

2904.8     187.1     193.5    1501.5 

22 

24 

2811.2      75.1     180.6     1449.9 

2908.1     187.5     193.9    1503.2 

24 

26 

2814.4      75.5    181.0     1451.7 

2911.3     188.0     194.4     1505.0 

26 

28 

2817.7      75.9    181.5    1453.5 

2914.5    188.4     194.8    1506.8 

28 

30 

2820.9      76.3     181.9    1455.2 

2917.7    188.8     195.3     1508-6 

30 

32 

2824.1       76.7     182.3     1457.0 

2921.0    189.2     195.7     1510.4 

32 

34 

28274       77.1     182.8     145S.8 

2924.2     189  7    196.2     1512.1 

34 

36 

2830.6      77.5     183.2     1460  6 

2927.4     190.1     196.7     1513.9 

36 

38 

2833.8      77.9     183.6    1462.3 

2930.6    190.5    197.1     1515.7 

38 

40 

2837.1      78.4    184.1     1464.1 

2933.9    190.9    197.6    1517.5 

40 

42 

28403      78.8    184.5    1465.9 

2937.1     191.4     198.0    1519.3 

42 

44 

2843  5    179.2    185.0    1467.7 

2940.3    191.9    198.5     1521.0 

44 

46 

2846.8     179.6    185.4     1409.5 

2943.5    192.4     198.9    1522.8 

46 

4S 

2850.0     180  0    1S5  9     1471.2 

2946.8    192.8    199.4     1524.6 

48 

50 

2853.2    180.4    186  3    1473  0 

2950  0    193.2    199.8    1526.4 

50 

K 

2856  5     180.8    186.8    1474.8 

2953.2    193.6    200.3    1528.2 

52 

54 

2859.7    181.2    187.2    1476.6 

2956.4    194.0    200.8     1530.0 

54 

56 

2862.9     181.6    187.6    1478.3 

2959.6    194.4    201.2    1531.7 

56 

58 

2866.2     1820    188.1     1480.1 

2962.9     194.8    201.7     1533.5 

5K 

60 

2869.4     182  5     188.5     1481  9 

2!  166.1     195.2    202.1     1535  3 

60 

276     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


30° 

31° 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

' 

0 

2966.1  195.2  202.1  1535.3 

3002.6  208.4  210  3  1589  0 

o 

2 

2909.3  195.6  202.6  1537.1 

3005.8  208.8  216.8  1590.H 

2 

4 

2972.5  190.1  203.1  ]  538.9 

3001).  0  209.3  217  2  1592.6 

4 

6 

2975.7  190.5  203.5  1540.7 

3072.2  209.7  217.7  1594.4 

6 

8 

2979.0  197.0  204.0  1542  5 

3075.4  210.2  2182  1596.2 

8 

10 

2HS2.2  197.4  204.5  1544  3 

3078.0  210.6  218  7  159S  i) 

10 

12 

2985.4  197.8  204.9  1546.0 

3081.8  211.1  219.2  1599.8 

12 

14 

2988.6  198.2  20o.4  1547.8 

3085.0  211.5  219.6  1601.0 

14 

16 

2991.8  198.6  205  9  1549.6 

3088.3  212.0  220!  1003.4 

16 

18 

29950  199.1  2003  1551.4 

3091.5  212.4  2»'0.6  1605.2 

18 

20 

2998.3  1995  206.8  1553.2 

3094.7  212.9  <>21  t  1607.0 

20 

2-2 

3001.5  199  9  207.3  1555.0 

3097.9  213.3  331.6  1608.8 

22 

24 

3004.7  200.4  207.7  1556.8 

3101.1  213.8  222  1  1610.0 

24 

26 

3007.9  200.8  208.2  1558.6 

3104.3  214.2  222.6  1612.4 

26 

28 

3011.1  201.3  208.7  1500.4 

3107.5  214.7  223.0  1614.2 

28 

30 

3014.3  201.7  209.1  1562.2 

3110.7  215.1  223.5  1616.0 

30 

32 

3017.6  2021  209.6  1504.0 

3113.9  215.0  224.0  1617.8 

32 

34 

3020.8  2026  210.1  1565.7 

3117.1  216.0  224.5  1619.6 

34 

36 

3024.0  203.0  210.5  1567.5 

3120.3  216.5  2^5.0  1021.4 

36 

38 

3027.2  203.5  211.0  1569.3 

3123.5  216.9  225.5  1623.2 

38 

40 

3030.4  203.9  211  5  1571.1 

3126.7  217.4  226.0  1625.0 

40 

42 

3033.6  204.3  2120  1572.9 

3129.9  217.8  226.5  1626.8 

42 

44 

3036.9  204.8  212.4  1574.7 

3133.1  218.3  227.0  1628.6 

44 

46 

30-10.1  205.2  212.9  1576.5 

3136.4  218.7  227.5  1030.5 

46 

48 

3043.3  205.7  213.4  1578.3 

3139.6  219.2  228.0  1632.3 

48 

50 

3046.5  206.1  213.9  1580.1 

3142.8  219.6  228.4  1634.1 

50 

52 

3049.7  206.5  214.4  1581.9 

3146.0  220.1  228.9  1635.9 

52 

54 

3052.9  207  0  214.8  1583.7 

3149.2  220.5  229.4  1637.7 

54 

56 

3056.2  207.4  215.3  1585.5 

3152.4  221.0  2J9.9  1639.5 

56 

58 

3059.4  207.9  215.8  1587.2 

3155.6  221.5  230.4  1641.3 

58 

60 

3062.6  208.4  216  3  1589.0 

3158.8  222.0  230.9  1643.1 

60 

32° 

33° 

L.  C.   M.    E.    T. 

L.  C    M.    E.    T. 

/ 

0 

3158.8  222.0  230.9  1643,1 

3254  9  236.0  246.1  1697  3 

0 

2 

3162.0  222  5  231.4  1644  9 

3258  1  230.4  246.6  1099.1 

2 

4 

3165.2  222.9  231.9  16467 

3261  3  236  9  247.1  1700  9 

4 

6 

3168.4  223.4  232.4  1048.5 

3204.5  2374  247.7  1702.7 

6 

8 

3171.6  223.8  232.9  1650.3 

3267  7  237.9  248.2  1704.5 

8 

10 

3174.8  224.3  233.4  1652  1 

3270.8  238.4  248  7  1706.4 

10 

12 

3178.0  224  8  233.9  1653.9 

3274  0  238  9  249.2  1708.2 

12 

14 

3181.2  225.2  234.4  1655  7 

3277  2  239  3  249.7  1710.0 

14 

16 

3184.4  225  7  234.9  1657  5 

3280  4  239.8  250.2  1711.8 

16 

18 

3187.6  226  1  235.4  1659.3 

3283.6  240.3  250.8  1713.6 

18 

20 

3190.8  226.6  2359  1661.1 

3286.8  240.8  251.3  1715.5 

20 

22 

3194.0  227.1  236.4  1662.9 

3290  0  241  2  2ol  8  1717.3 

22 

24 

3197  2  227.5  236  9  1664.7 

3293.2  241  7  252.3  1719.1 

24 

20 

3200.4  228.0  237.4  160(5  5 

3296.4  242.2  252  9  1720.9 

26 

28 

3203.6  228.4  2379  10683 

3299.6  242,7  253  4  1720.7 

28 

30 

3206.8  228  9  238.4  1670.1 

3302  7  243  2  253.9  1724.6 

30 

32 

3210.0  229.4  239.0  1671.9 

3305  9  243.6  254.4  1726.4 

32 

34 

3213.2  229  8  239.5  1673  7 

3309  1  244.1  255  0  1728.2 

34 

36 

3216.5  230.3  240.0  1675  5 

3312  3  244.6  255.5  1730.0 

36 

38 

3219.7  230.7  240.5  1677.4 

3315  5  245.1  256.0  1731.8 

38 

40 

3222.9  231  2  241.0  1679.2 

3318.7  245.6  256  5  1733.6 

40 

42 

3226  1  231.7  241  5  1681.0 

3321.9  246  0  257  1  1735.5 

42 

44 

3629.3  232.2  242.0  1682  8 

3325.1  240.5  257.6  1737.3 

44 

46 

3230.5  232.6  242.5  1684.6 

3328.3  247.0  258.1  1739.1 

46 

48 

3235.7  233.1  243.0  1686  4 

3331.5  247  5  258  6   740.9 

48 

50 

3,  '38.  9  233.5  243.5  16882 

3334.6  248.0  259  2   742.7 

50 

52 

3242.1  234.0  244  1  1690  0 

3337.8  248  4  259.7   744.6 

52 

54 

3245  3  234.5  244  6  1691.8 

3341.0  248.9  260  2   746.4 

54 

56 

3248.5  235  0  245.1  Ki<)3  7 

3344-2  2)9.4  260  8   74S.2 

56 

58 

3251.7  235.5  245.0  1  (•>'.):>  r> 

3347.4  249.9  261.3   750.0 

58 

60 

3254  9  236  0  246  1  1097  3 

3350-6  250.4  261  8   751.8 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.  2-7 


34° 

85° 

' 

L.  C.        M.         E.         T. 

L.C.        M.          E.          T. 

0 

3350.6    250.4    261.8    1751.8 

3446.1     265.2    278.1     1806.7 

0 

2 

3353.8    250.8    202.3     1753.7 

3449.3    265.7    278.6    1808.5 

2 

4 

3357.0    251.2    2G2.9    1755.5 

3452.5    266.2    279.2    1810.3 

4 

6 

3360.1     251.7    263.4    1757.3 

3455.6    266.7    279.7    1812.2 

6 

8 

3363.3    252.2    204.0    1759.1 

3458.8    267.2    280.3    1814.0 

8 

10 

3366.5    252.7    264.5    1761.0 

3462.0    267.7    280.8     1815.8 

10 

12 

8309.  7    253.2    265.0      762.8 

3465.2    268.2    281.4     1817.7 

12 

14 

3372.9    253.7    265.6      764.6 

3468.3    268.7    2S1.9    1819.5 

14 

16 

3370.1     254.2    266.1      706.  4 

3471.5    269.2    282.5    1821.3 

16 

18 

3379.2    254.7    266.7      708.3 

3474.7    269.7    283.0    1823.2 

18 

20 

33^2.4    255.2    267.2      770.1 

34'<7.9    270.2    283.6    1825.0 

20 

22 

3385.6    255.7    267.7      771.9 

34S1.0    270.7    284.2    1826.8 

22 

24 

3388.8    256.2    268,3      773.7 

3484.2    271.2    284.7    1828.7 

24 

26 

3:592.0    256.7    268.8      775.6 

3487.4    271.7    285.3    1830.5 

26 

28 

3395.2    257.2    269.3      777  4 

3490.6    272.2    285.9    1832.3 

28 

30 

3398.3    257.7    269.9      779.2 

3493.7    272.7    286.4     1834.2 

30 

3! 

3401.5    258.2    270.4      781.0 

3496.9    273.2    287.0    1836.0 

32 

34 

3404.7    258.7    271.0      782.9 

3500.1     273.  V    287.5    1837.8 

34 

36 

3407.9    2592    271.5      784.7 

3503.3    274.2    8SS.1     1839.7 

36 

38 

3411.1     259.7    2720    1786.5 

3506.5    274.7    288.7    1841.5 

38 

40 

3414.3    260.2    272.6    1788.4 

3509.6    275.2    289.2    1843.4 

40 

42 

3417*4    260.7    273.1     1790.2 

3512.8    275.7    289.8    1845.2 

42 

44 

3120.6    261.2    273.7     1792.0 

3516.0    276  2    290.4     1847.1 

44 

46 

34-J3.8    261.7    274.2    17939 

3519.2    276.7    290.9    1848.9 

46 

48 

3427.0    262.2    274.8    1795.7 

3522.3    277.2    291.5     1850.7 

48 

50 

3430.2    262.7    275.3    1797.5 

3525.5    277.7    292.0     1852.6 

50 

52 

3433.4    263.2    275.9      799.3 

3528.7    278.2    292.6    1854.4 

52 

54 

3436.5    263.7    276.4    1801.2 

3531.9    278.7    293.2    1856.3 

54 

56 

3439.7    264.2    277.0    1803.0 

3535.0    279.2    293.7    1858.1 

56 

58 

3442.9    264.7    277.5    1804.8 

3538.2    279.8    294.3     1859.9 

58 

GO 

3446  1     265.2    278.1     1806.7 

3541.4    280.4    294.9    1861.8 

60 

86° 

87° 

L.  C.        M.         E.          T. 

L.  C.        M.         E.          T. 

t 

0 

3541.4    280.4    294.9    1861.8 

3036.3    296.1     312.3     1917.3 

0 

2 

3544.6    280  9    295.4    1863.6 

3639.5    296.6    312.8    1919.1 

2 

4 

3547.7    281.4    290.0    1865.5 

3642  6    297  1     313.4    1921.0 

4 

G 

3550.9    281.9    290.6    1867.3 

3645.8    297.7    314.0    1922.8 

6 

8 

3554.0    282.5    297.2    1869.2 

3648.9    298.2    314.6    1924.7 

8 

10 

3557.2    283.0    297.7    1871.0 

3652.1     298.7    315.2    1926.5 

10 

12 

3560.4    283.5    298.3    1872.9 

3655.2    299.3    315.8     1928.4 

12 

14 

3563.5    284.0    298.9    1874.7 

3658.4    299.8    316.4     1930  2 

14 

16 

3566.7    284.6    299.5    1876.5 

3661.6    300.3    317.0    1932.1 

16 

18 

3569.9    285.1     300.0    1878.4 

3064.  7    300.9    317.5    1933.9 

18 

20 

3573.0    285.6    300.6    1880.2 

3667.9    301.4    318.1     1935.8 

20 

22 

3576.2    286.1     301.2    1882.1 

3671.0    301.9    318.7    1937.6 

22 

24 

3579  4    286.7    301.8    1883.9 

3674.2    302.5    319.3     1939.5 

24 

26 

35S2.5    287.2    302.3    18S5.8 

81577.3    303.0    319.9    1941.3 

26 

28 

3585.7    287.7    302.9    1887.6 

3080.5    303.5    320  5    1943.2 

28 

30 

3588.8    288.2    303.5    1889.5 

3683.6    304.1     321.1     1945.0 

30 

32 

3592.0    288.8    304.1     1891.3 

3680.8    304.6    321.7     1946.9 

32 

34 

3595.2    289.3    304.6     181)3.2 

3690.0    305.1     322.3     1948.8 

34 

36 

3598.  3    289  8    305.2    18115.0 

36'.):!.  1     305.7    322.9    1950.6 

36 

38 

3601.5    290.3    305.8     1896.9 

3690.3    306.2    323.5     1952.5 

38 

40 

3604.7    2909    306.4    1898.7 

3699.4    306.7    324.2    1954.4 

40 

42 

3607.8    291.4    307.0    1900.6 

3702.6    307.3    324.8     1956.2 

42 

44 

3611.0    291.9    307.5    1902  4 

3705.7  ,307.8    325.4    19:,S.l 

44 

46 

3614.1     292.4    308.1     1904.3 

3708.9    308.3    326.0    1960.0 

46 

48 

3017.3    293.0    308.7    1906  .1 

3712.1     308.9    326  6    1961.8 

48 

50 

3620.5    293.5    309.3    1908.0 

3715.2    309.4    327  2    1963.7 

50 

52 

3623  6    294.0    309.9     1909.8 

371  S.  4    309  9    327.8     1%5.5 

52 

54 

3020.  S     294.5     310.5     1011.7 

3721   5     310.5    328.4     1967.4 

54 

58 

3630.0    295.1     311.1     1913.5 

3724.7    311.0    3:29.0     1909.3 

56 

58 

3633.1     295.6    311.7    1915.4 

3727.8    311.6     329.6     1971   1 

58 

60 

3036.3     296.1     312.3    1917  3 

3731.0     312.2    3302     1973.0 

GO 

278     IX.— FUNCTIONS  OF  A  ONE  DEGREE  CURVE. 


38° 

89° 

L.  C.    M.    E.     T. 

L.  C    M    E.    T. 

0 

3731.0  312.2  330.2  1973.0 

38255  328-7  348.7  2029.1 

0 

2 

8734.1  312  7  330.8  1974.9 

3828.6  329  ..2  349  3  2031.0 

o 

4 

3737.3  3133  331.4  1976-7 

3831.8  329.8  349.9  2032  9 

4 

6 

37404  313.8  332.0  1978.6 

3834.9  330-3  350.6  2034  7 

6 

8 

3743.6  314.4  332.6  1980-5 

3S38.0  330.9  351.2  2036.6 

8 

10 

3746.7  314.9  333.2  1982.3 

3841.2  331.5  351.8  2038  5 

10 

12 

3749.9  315.5  333.8  1984.2 

3844  3  332.0  352-4  2040.4 

12 

14 

3753  0  316  0  334.5  1986.1 

3847.4  332.6  353.1  2042  3 

14 

16 

3756  2  316.6  335.1  1987.9 

3850.6  333.2  353.7  2044.1 

16 

18 

3759.3  317.1  335.7  1989.8 

3853  7  333  7  354.3  20J6.0 

18 

20 

37625  3177  3363  1991.7 

3856.8  334.3  354.9  2047.9 

20 

22 

3765.6  318  2  336  9  1993-6 

3860  0  334.9  355.6  2049.8 

22 

24 

3768.8  318.8  337.5  1995.4 

3863.1  335.4  356.2  2051.7 

24 

26 

3771  9  319.3  338.1  1997.3 

3866  2  3:56.0  356.9  2053  5 

26 

28 

3775.1  319.9  338.7  1999.2 

3869.4  336.6  357  5  2055.4 

28 

30 

3778.2  320  4  339.4  2001.0 

38725  337.1  358.1  2057-3 

30 

32 

3781.4  321.0  340.0  2002  9 

3875  6  337  7  358.8  2059.2 

32 

34 

3784.5  321.5  340-6  2004.8 

3878.8  338.3  359  4  2061.1 

34 

36 

3787.7  322.1  341.2  2006.6 

3881.9  338.8  360.1  2063.0 

36 

38 

3790.8  322.6  341.8  2008.5 

3885.0  339  4  360.7  2064-8 

38 

40 

3794.0  3232  342.4  20104 

3888.2  340.0  361.3  2066.7 

40 

42 

37971  323.7  343.1  20123 

3891.3  MO.  5  362.0  2068  6 

42 

44 

3800.3  324.3  343.7  2014.1 

3894.4  341  1  362.6  2070.5 

44 

46 

3803.4  324.8  344.3  2016  0 

3897.6  341.7  363.3  2072.4 

46 

48 

3806.6  325.4  344.9  2017.9 

3900  7  342  2  363.9  :.'074  2 

48 

50 

3809.7  325  9  345.6  2019.7 

3903.8  342  8  364.5  2076  1 

50 

52 

3812.9  3265  346.2  2021.6 

3907  0  343  4  365.2  2078.0 

52 

54 

3816.0  3270  346.8  2023.5 

3910.1  343.9  365.8  20799 

54 

56 

3819.2  327.6  347  4  2025  4 

3913.2  344.5  366.5  2081.8 

56 

68 

3822.3  328.1  348  1  2027.2 

3916  4  345  1  367  1  2083.7 

58 

60 

3825.5  328.7  348-7  2029.1 

3919  5  345.6  367  7  2085-5 

60 

40" 

41° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.    T. 

0 

3919.5  345.6  367.7  2085.5 

4013.4  362.9  387.4  2142.3 

0 

2 

:i(.)2.'.(>  346.1  368.4  20S7  4 

4016.5  ::63.4  388.1  2144.2 

2 

4 

3925.8  346  7  369  0  2089  3 

4019.6  364.0  388.8  2146.1 

4 

6 

3928  9  347.2  369.7  2091  2 

4022.7  364  5  389.4  2148.0 

6 

8 

3932  0  347  8  370.3  20V3.  1 

4025.9  365.1  390  1  2149.9 

8 

10 

3935.1  348.4  371.0  2095  0 

4029.0  365.6  390.7  2151.9 

10 

12 

3938.3  348.9  371  6  2096  9 

4032.1  366.2  391.4  2153.8 

12 

14 

3941.4  349.5  372.3  209S.S 

4035.2  366.8  392.1  2155.7 

14 

16 

3944.5  3501  372.9  2100.7 

4038.3  367.4  392.7  2157.6 

16 

18 

3947.7  3507  373.6  21026 

4041.4  368.0  393.4  2159.5 

18 

20 

3950.8  351.3  374.3  2104.5 

4044.6  368.6  394.1  2161.4 

20 

22 

3953.9  351  8  374.9  2100.3 

4047.7  369.2  394.7  2163.3 

22 

24 

3957.1  352.4  375.6  2108.2 

4050.8  369.8  395.4  2165.2 

24 

20 

3960.2  353.0  376.2  2110.1 

40539  37'0.4  396.1  2167.1 

26 

28 

39153.3  353  6  376.9  2112  0 

4057.0  371.0  396.8  2169.0 

28 

30 

39G6.4  354.2  377.5  2113  9 

40ii0.1  371.6  397.5  2170  9 

30 

33 

3969.6  354.7  378.2  2115  8 

4063.3  372.2  398.1  2172.8 

32 

34 

3972.7  355.3  378.8  2117  7 

4066.4  37'2.8  39S.8  2174.7 

34 

30 

3H75.8  355.9  379.5  2119  6 

K  169.  5  373.4  399.5  2176.6 

36 

38 

3979.0  356.5  380  1  2121.5 

4072.6  374.0  400.2  2178.5 

38 

40 

3982.1  357.1  380.8  2123.4 

4075.7  374.6  400.9  2180.4 

40 

42 

3985.2  357  6  381.4  2125  3 

4078.8  375.2  401.5  2182.4 

42 

44 

3988.4  358.2  382.1  SI  27.  2 

4082.  0  375  8  402.2  21  SI.  3 

44 

46 

:W'.M  5  358  8  382.8  2129.1 

4085.1  376.4  402.9  2186.2 

46 

48 

3994.6  359.4  383.4  2131.0 

4088.2  377.0  403.6  2188.1 

48 

50 

3997  7  3GO  0  384.1  2132.9 

4091.3  377.6  404.3  2190.0 

50 

52 

4000  9  360  5  384.8  2134.7 

4094.4  378.2  404.9  2191.9 

52 

54 

4004.0  361  1  385.4  2136.6 

4097.5  378.8  405.6  2193.8 

54 

56 

4007  1  361.7  386.1  2  13*.  5 

4100.7  379.4  406.3  2195.7 

56 

58 

4010  3  362.3  386  8  2  NO  4 

4103.8  3800  407.0  2197.6 

58 

60 

4013  4  362.9  387  4  2142.3 

4106.9  380.6  407.7  2199.5 

60 

IX.     FUNCTIONS   OF   A   ONE-DEGREE   CURVE.     279 


4!>° 

4:5° 

L.  C.   M.   E.     T. 

L.  C.    M.    E.     T. 

0 

4106.9  380.6  407.7  2199.5 

4200.1  398.7  428.6  2257.1 

0 

2 

4110.0  381.2  408.  8  2201.4 

4203.2  399.3  429.3  2259.0 

2 

4 

4113  1  381.8  409.0  2203.3 

4206.3  399.9  430.0  2261.0 

4 

6 

4116.2  382.4  409.7  2205.3 

4209.4  400.5  430.7  2262.9 

6 

8 

4119.3  383.0  410.4  2207.2 

4212.5  401.1  431.4  2264.8 

8 

10 

4122.4  383.6  411.1  2209.1 

4215.6  401.7  432.1  2266.7 

10 

12 

4125.5  384.  2  411.8  2211.0 

4218.7  402.4  432.8  2268.7 

12 

14 

4128.6  384.8  412.5  2212.9 

4221.8  403.0  433.5  2270.6 

14 

16 

4131.8  3854  413.2  2214.9 

4224.9  403.6  434.2  2272.5 

16 

18 

4134.9  386.0  413.9  2216.8 

4228.0  404.2  434.9  2274.5 

18 

20 

4138  0  380.6  414.6  2218.7 

4231.1  404.8  435.6  2276.4 

20 

4141.1  387.2  415.3  2220.6 

4234.2  405.4  436.3  2278.3 

22 

24 

4144.2  387.8  416.0  2222.5 

4237.3  406.1  437.0  2280.2 

24 

26 

4147.3  388.4  416.6  2224.4 

4210.4  406.7  437.8  2282.2 

26 

28 

4150.4  389.0  417.3  2226.4 

4243.5  407.3  438.5  2284.1 

28 

30 

4153.5  389.  6  418.0  2228.3 

4246.5  407.9  439.2  2286.0 

30 

32 

4156.6  390,2  418.7  2230.2 

4249.6  408.5  439.9  2288.0 

32 

34 

4159.7  390.8  419.4  2232.1 

4252.7  409.1  440.6  2289.9 

34 

30 

4162.8  391.4  420.1  2234.0 

4255.8  409.8  441.4  2291.8 

36 

38 

4165.9  392.0  420.8  2236.0 

4258.9  410.4  442.1  2293.8 

38 

40 

4169.0  392.6  421.5  2237.9 

4262.0  411.0  442.8  2295.7 

40 

42 

4172.1  393.2  422.2  2239.8 

4265.1  411.6  443.5  2297.7 

42 

44 

4175.2  393.8  422.9  2241.7 

4268.2  412.2  444.2  2299.6 

44 

46 

4178.4  394.4  423.6  2243.6 

4271.3  412.8  445.0  2301.5 

46 

48 

4181.5  395.0  424.3  2245.6 

4274.4  413.5  445.7  2303.5 

48 

50 

4184.6  395.6  425.0  2247.5 

4277.5  414.1  446.4  2305.4 

50 

52 

4187.7  396.2  425.7  2249.4 

4280.6  414.7  447.1  2307.3 

52 

54 

4190.8  396.8  426.4  2251.3 

4283.7  415.3  447.8  2309.3 

54 

56 

4193.9  397  4  427.1  2253.3 

42S6.8  415.9  448.6  2311.2 

56 

58 

4197.0  398.0  427.8  2255.2 

4289.9  416.5  449.3  2313.1 

58 

60 

4-200.1  398.7  428.6  2257.1 

4293.0  417.2  450.0  2315.1 

60 

44° 

45° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

4293.0  417.2  450.0  2315.1 

4385.5  436.2  472.1  2373.4 

0 

2 

4296.1  417.8  450.7  2317.0 

4388.6  436.8  472.9  2375.4 

2 

4 

4299.2  418.4  451.5  2319.0 

4391.7  437.5  473.6  2377.3 

4 

6 

4302.2  419.1  452.2  2320.9 

4394.7  438.1  474.4  2379.3 

6 

8 

4305.3  419.7  452  9  2322.8 

4397.8  438.8  475.1  2381.2 

8 

10 

4308.4  420.3  453.7  2324.8 

4400.9  439.4  475.9  2383.2 

10 

12 

4311.5  421.0  454.4  2326.7 

4404.0  440.0  476  6  2385.2 

12 

14 

4314.6  421.6  455.1  2328.7 

4407.0  440.7  477.4  2387.1 

14 

16 

4317.7  422.2  455.9  2:330.6 

4410.1  441.3  478.1  2389.1 

16 

18 

4320.7  422.9  4566  2332.6 

4413.2  442.0  478.9  2391.0 

18 

20 

4323.8  423.5  457.3  2334.5 

4416.3  442.6  479.6  2393.0 

20 

22 

4326.9  424.1  458.1  2336.4 

4419.3  443.2  480.4  2394.9 

22 

24 

4330.0  424  .8  458.8  2338.4 

4422.4  443.9  481.1  2396.9 

24 

26 

4333.1  425  4  459.5  2340  3 

4425.5  444.5  481.9  2398.8 

26 

28 

4336  2  426.0  460  3  2342.3 

4428.6  445.2  482.6  2400.8 

28 

30 

4339.2  426.7  461  0  2344.2 

4431.6  445.8  483.4  2402.8 

30 

32 

4342.3  427.3  461.7  2346.1 

443J.7  446.4  484.2  2404.7 

32 

34 

4345.4  427  9  462.5  2348.1 

4437.8  447.1  484.9  2406.7 

34 

36 

4348.5  4286  463.2  2350.0 

4440.9  447.7  485.7  2408.6 

36 

38 

4351.6  429.2  463.9  2352.0 

4444.0  448.3  486.5  2410.6 

38 

40 

4354.7  429.8  464.1  2353.9 

4447.0  448.9  487.2  2412.6 

40 

42 

4357.7  430.5  465.4  2355.9 

4450.1  449.5  488.0  2414.5 

42 

44 

4360.8  431.1  466.2  2357.8 

4453  2  450.2  488.7  2416.5 

44 

46 

4303.9  431.7  466.9  235!).  8 

4456.3  450.8  489.5  2418.5 

46 

48 

4367.0  432.4  467.7  2361.7 

4459.3  451  5  490.3  2420.4 

48 

50 

4370.1  433.0  4684  2363.7 

4462.4  452.1  491.0  2422.4 

50 

52 

4373.2  433.6  469.1  2365.6 

44655  452.7  491.8  2424.4 

52 

54 

4376.2  434.3  469.9  2367.6 

4468.6  453  4  492.5  2426.3 

54 

5ti 

43~9.3  434.9  470.6  23G9.5 

4471.6  454.1  493.3  24283 

56 

f>» 

4W2.4  435  b  471.4  23^1.5 

4474.7  4*4.8  494.1  24302 

58 

01* 

438.7  5  480.2  472.1  2373  4 

4477  8  455.5  494.8  2432.2 

60 

280     IX.— FUNCTIONS  OP  A  ONE  DEGREE  CURVE. 


46° 

47° 

L.  C.   M.    E.    T. 

L  C.   M.    E.    T. 

' 

0 

4477.8  455.5  494.8  2432.2 

4569.7  475.2  518.3  2491.5 

0 

2 

4480.9  456.1  495.6  2434.2 

4572.7  475.9  519.0  2493.4 

2 

4 

4483.9  456.8  490.5  2436.1 

4575.8  470.5  519.8  2495.4 

4 

6 

4487.0  457.4  497.2  2438.1 

4578.8  477.2  520.6  2497.4 

6 

8 

4490.0  458.1  497.9  2440.1 

4581.9  477.8  521.4  2499.4 

8 

10 

4493.1  458.7  498.7  2442.1 

4584.9  478.5  522.2  2501.4 

10 

12 

4496.2  459.4  499.5  2444.0 

4588.0  479.2  523.0  2503.4 

12 

14 

4499.2  460.0  500.3  2446.0 

4591.0  479.8  523.8  2505.4 

14 

16 

4502.3  460.7  501.0  2448.0 

4594.1  480.5  524.6  2507.3 

16 

18 

4505.4  461.3  501.8  2449.9 

4597.1  481.1  525.4  2509.3 

18 

20 

4508.4  462.0  502.6  2451.9 

4600.2  481.7  526.2  2511.3 

20 

22 

4511.5  462.7  503.4  2453.9 

4603.2  482.3  527.0  2513.3 

22 

24 

4514.6  463.3  504.1  2455,9 

4606.3  483.0  527.8  2515.3 

24 

26 

4517.6  464.0  504.9  2457.8 

4609.3  483.7  528.6  2517.3 

26 

28 

4520.7  464.6  505.7  2459.8 

4612.4  484.3  529.4  2519.3 

28 

30 

4523.7  405.  3  506.5  2461.8 

4615.4  485.0  530.2  2521.2 

30 

32 

4526.3  466.0  507.3  2463.  S 

4618.5  485.7  531.0  2523.2 

32 

34 

4529.9  466.6  508.0  2465.7 

4621.5  486.3  531.8  2525.2 

34 

36 

4532.9  467.3  508.8  2467.7 

4624.6  487.0  532.6  2527.2 

36 

88 

4536.0  467.9  509.6  2469.7 

4627.6  487.7  533.4  2529.2 

38 

40 

4539.1  468.6  510.4  2471.7 

4630.7  488.4  534.2  2531.2 

40 

42 

4542.1  469.3  511.1  2473.6 

4633.7  489.1  535.0  2533.2 

42 

44 

4545.2  469.9  511.9  2475.6 

4636.8  489.8  535.8  2535.2 

44 

46 

4548.2  470.6  512.7  2477.6 

4639.8  490.5  536.6  2537.2 

46 

48 

4551.3  471.2  513.5  2479.6 

4642.9  491.2  537.4  2539.2 

48 

50 

4554.4  471.9  514.3  2481.6 

4645.9  491.9  538.2  2541.2 

50 

52 

4557.4  472.6  515.1  2483.5 

4649.0  492.6  539.0  2543.1 

52 

54 

4560.5  473.2  515.9  2485.5 

4652.0  493.3  539.8  2545.1 

54 

56 

4563.6  473.9  516.7  2487.5 

4655.1  494.0  540.6  2547.1 

56 

58 

4566.6  474.5  517.5  2489.5 

4658.1  494.7  541.4  2549.1 

58 

60 

4569.7  475.2  518.3  2491.5 

4661.2  495.4  542.3  2551.1 

60 

48° 

49° 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

f 

0 

4661.2  495.4  542.3  2551.1 

4752.3  515.9  567.0  2611.3 

0 

2 

4664.2  496.0  543.1  2553.1 

4755.3  510.5  567.8  2613.3 

2 

4 

4667.3  496.7  543.9  2555.1 

4758.4  517.2  568.7  2615.3 

4 

6 

4670.3  497.4  544.7  2557.1 

4761.4  517.9  569.5  2617.3 

6 

8 

4673.3  498.1  545.5  2559.1 

4764.4  518.6  570.3  2619.3 

8 

10 

4676.4  498.8  546.4  25G1.1 

4767.4  519.3  571.2  2621.4 

10 

12 

4679.4  499.4  547.2  2563. 

4770.5  520.0  572.0  2023.4 

12 

14 

4682.5  500.1  548.0  2565. 

4773.5  520.7  572.8  2625.4 

14 

16 

4685.5  500.8  548.8  2567. 

4776.5  521.4  573.7  2627.4 

16 

18 

4688.5  501.5  549.6  2569. 

4779.6  522.1  574.5  2629.4 

18 

20 

4691.6  502.2  550.5  2571. 

4782.6  522.8  575.3  2631.4 

20 

22 

4694.6  502.8  551.3  2573. 

4785.6  523.5  576.2  2633.5 

22 

24 

4697.6  503.5  552.1  2575. 

4788.7  524.2  577.0  2635.5 

24 

26 

4700.7  504.2  552.9  2577. 

4791.7  524.9  577.9  2637.5 

26 

28 

4703.7  504.9  553.7  2579. 

4791.7  525.6  578.7  2639.5 

28 

30 

4706.7  505.6  554.6  2581. 

4797.7  526.3  579.6  2641.5 

30 

32 

4709.8  506.2  555.4  2583. 

4800.8  527.0  580.4  2043.5 

32 

34 

4712.8  506.9  556.2  2585.1 

4803.8  527.7  581.3  2645.6 

34 

36 

4715.9  507.6  557.0  2587.2 

4806.8  528.4  582.1  2047.6 

36 

38 

4718.9  508.3  557.8  2589.2 

4809.9  529.1  583.0  2649.6 

38 

40 

4721.9  509.0  558.7  2591.2 

4812.9  529.8  583.8  2651.6 

40 

42 

4725.0  509.6  550.5  2593.2 

4815.9  530.5  584.7  2653.7 

42 

44 

4728.0  510.3  560.3  2595.2 

4819.0  531.2  585.5  2655.7 

44 

46 

4731.0  511.0  561.2  2597.2 

4822.0  531.9  586.4  2657.7 

46 

48 

4734.1  511.7  562.0  2599.2 

4825.0  532.6  587.2  2659.7 

48 

50 

4737.1  512.4  562.8  2601.2 

4828.0  533.3  588.1  2661.8 

50 

52 

4740.2  513.1  563.7  2603.2 

4831.1  534.0  588.9  2603.8 

52 

54 

4743.2  513.8  564.5  2(505.2 

4834.1  534.7  589.8  2(105.8 

54 

56 

4746.2  514.5  565.3  2007.2 

4837.1  535.4  590.6  2607.8 

56 

58 

474'.».3  515.2  566.2  2609.3 

4840.2  530.1  591.5  2069.9 

58 

60 

4752.3  515.9  567.0  2611.3 

4843.2  530.8  592.4  2671.9 

60 

IX.— FUNCTIONS  OF   A  ONE-DEGREE  CURVE.     281 


50° 

51° 

L.  C.   M.   E.    T. 

L.  C.    M.    E.    T. 

0 

4843.2  536.8  592.4  2671.9 

4933  6  558.2  618.5  2733.0 

0 

o 

4816.2  537.5  593.2  2673  9 

4936.6  558.9  619.3  2735.1 

2 

4 

4849.2  538.2  594.1  2676.0 

4939.6  559.7  620.2  2737.1 

4 

6 

4852.2  538.9  594.9  2678.0 

4942.6  560.4  621.1  2739.2 

6 

8 

4855.2  539.6  595.8  2680.0 

4945.6  561.1  622.0  2741.2 

8 

10 

4858.3  540.3  596.7  2682.1 

4948.6  561.8  622.9  2743.3 

10 

18 

4861.3  541.0  597.5  2684.1 

4951.6  562.5  623.7  2745.3 

12 

14 

4864.3  541.7  598.4  2686.1 

4954.6  563.3  624.6  2747.4 

14 

16 

4867.3  542.4  599.3  2688.2 

4957.6  584.0  625.5  2749.4 

16 

18 

4870.3  543.1  600.1  2690.2 

4960.6  564.7  626.4  2751.5 

18 

20 

4873.3  543.9  601.0  2692.3 

4963.6  565.4  627.3  2753.5 

20 

22 

4876.3  544.6  601.9  2694.3 

4966.6  5(56.2  628.2  2755.6 

22 

4879.4  545.3  602.7  2696.3 

4969.6  566.9  629.9  2757.7 

24 

26 

4882.4  546.0  603.6  2698.4 

4972.6  567.6  630.0  2759.7 

26 

28 

4885.4  546.7  604.5  2700.4 

4975.6  568.3  630.9  2761.8 

28 

30 

4888.4  547.4  605.3  2702.4 

4978.6  569.1  631.8  2763.8 

30 

32 

4891.4  548.1  606.2  2704.5 

4981.6  569.8  632.7  2765.9 

32 

34 

4894.4  548.8  607.0  2706.5 

4984.6  570.5  633.6  2767.9 

34 

36 

4897.4  549.5  607.9  2708.6 

4987.7  571.2  634.5  2770.0 

36 

38 

4900.4  550.2  608.8  2710.6 

4990.7  572.0  635.3  2772.0 

38 

40 

4903.5  551.0  609.7  2712.6 

4993.7  572.7  636.2  2774.1 

40 

42 

4906.5  551.7  610.5  2714.7 

4996.7  573.4  637.1  2776.2 

42 

44 

4909.5  552.4  611.4  2716.7 

4999.   574.1  638.0  2778.2 

44 

46 

4912.5  553.1  612.3  2718.8 

5002.   574.9  638.9  2780.3 

46 

48 

4915.5  553.8  613.2  2720.8 

5005.   575.6  639.8  2782.3 

48 

50 

4918.5  554.5  614.1  2722.8 

5008.   576.3  640.7  2784.4 

50 

52 

4921.5  555.2  614.9  2724.9 

5011.   577.0  641.6  2786.4 

52 

54 

4924.6  555.9  615.8  2726.9 

5014.   577.8  642.5  2788.5 

54 

56 

4927.6  556.6  616.7  2729.0 

5017.   578.5  643.4  2790.6 

56 

58 

4930.6  557.4  617,6  2731.0 

5020.7  579.2  644.3  2792.6 

58 

60 

4933.6  558.2  618.5  2733.0 

5023.7  579.9  645.2  2794.7 

60 

52° 

53° 

L.  C.   M.   E.     T. 

L.  C.   M.   E.     T. 

0 

5023.7  579.9  645.2  2794.7 

5113.5  602.0  672.7  2856.9 

0 

2 

5026.7  580.6  6-16.1  2796.8 

5116.5  602.8  673.7  2858.9 

2 

4 

5029.7  581.3  647.0  2798.8 

5119.4  603.5  674.6  2861.0 

4 

6 

5032.7  582.1  647.9  2800.9 

5122.4  604.3  675.5  2863.1 

6 

8 

5035.7  582.8  648.9  2803.0 

5125.4  605.0  676.4  2865.2 

8 

10 

5038.7  583.5  649.8  2805.0 

5128.4  605.8  677.4  2867.3 

10 

12 

5041.7  584.3  650.7  2807.1 

5131.3  606.5  678.3  2869.4 

12 

14 

5044.7  585.0  651.6  2809.2 

5134.3  607.3  679.2  2871.5 

14 

16 

5047.7  585.7  652.5  2811.2 

5137.3  608.0  680.2  2873.5 

16 

18 

5050.7  586.5  653.4  2813.3 

5140.3  608.8  681.1  2875.6 

18 

20 

5053.6  587.2  654.3  2815.4 

5143.2  609.5  682.0  2877.7 

20 

22 

5056.6  587.9  655.2  2817.4 

5146.2  610.3  683.0  2879.8 

22 

24 

5059.6  588.7  656.2  2819.5 

5149.2  611.0  683.9  2881.9 

24 

26 

5062.6  589.4  657.1  2821.6 

5152.1  611.8  684.9  2884.0 

26 

28 

5065.6  590.1  658.0  2823.6 

5155.1  612.5  685.8  2886.1 

28 

30 

5068.6  590.9  658.9  2825.7 

5158.1  613.3  686.7  2888.1 

30 

32 

5071.6  591.6  659.8  2827.8 

5161.   614.0  687.7  2890.2 

32 

34 

5074.6  592.3  660.7  2829.8 

5164.0  614.8  688.6  2892.3 

34 

36 

5077.6  593.1  661.6  2831.9 

5167.0  615.5  689.6  2894.4 

36 

38 

5080.6  593.8  662.5  2834.0 

5170.0  616.3  690.5  2896.5 

38 

40 

5083.6  594.5  663.5  2836.1 

5173.0  617.0  691.5  2898.6 

40 

42 

5086.6  595.3  664.4  2838.2 

5175.9  617.8  602.4  2900.7 

42 

44 

5089.6  596.0  665.3  2840.2 

5178.9  61S.5  693.4  2902.8 

44 

46 

5092.6  596.7  666.  2  2842.3 

51S1.9  619.3  094.3  2904.9 

46 

48 

5095.6  597.5  667.2  2844.4 

5184.9  620.1  695.3  2907.0 

48 

50 

5098.6  598.2  668.1  2846.5 

51t>7.8  620.8  690.2  2909.1 

50 

52 

5101.6  598.9  669.0  2848.5 

5190.8  621.5  697.1  2911.2 

52 

54 

5104.6  599.7  669.9  2850.6 

5193.8  622.3  698.1  2913.3 

54 

66 

5107.6  600.4  670.9  2S.V.'.7 

5196.7  (523.  0  <>««).0  2915.4 

56 

58 

5110.6  601.2  671.  S  -JSr.J.S 

5199.7  623.8  700.0  2917.5 

58 

60 

5113.5  602.0  672.7  281G.9 

520.  '.7  624.6  700.9  2919.5 

60 

282      IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


54° 

55° 

L.  C.       M.         E.          T. 

L.  C.       M.          E.          T. 

' 

0 

5202.7    G2I.6    700.9    2919.5 

5291.7    647.4    729.9    2982.8 

0 

2 

5205.7    025.4    701.9    2921.6 

5294.6    648.1     730.9    2984.9 

4 

5208.0    020.1     702.8    2923.8 

5297.6    0^8.9    131  9    2987.1 

4 

6 

5211.6    626.9    7038    2925.9 

53005    049.0    732.9    2989.2 

6 

8 

5214.6    027.6    704.8    8928.0 

5303.5    0504    733.8    2991.3 

8 

10 

5217.5    628.4    705.7    2930.1 

5300  4    051.2    734.8    2993.4 

10 

12 

5220.5    629.2    700.7    2932.2 

5309.4    052.0    735.8    2995.5 

12 

14 

5223.5    629.9    707.7    2934.3 

5312.3    052.7    730.8    2997.7 

14 

10 

5226.4    030.7    708-6    2936.4 

5315.3    653.5    737.8    2999.8 

16 

re 

5229.4    031.4    709.6    2938.5 

5318.2    054.3    738.7    3001.9 

18 

2X1 

5232.4    632.2    710.5    2940.6 

5321.2    655.1     739.  "    3004.0 

20 

*> 

5235.3    633.0    711.5    2942.7 

5324.1     655.8     740."    3000.2 

22 

24 

5238.3    633.7    712.5    2944.8 

5327.1     656.6    741."    3008.3 

2t 

20 

5241.3    634.5    713.4    2940.9 

f>330.0    057.4    742."    3010.4 

26 

28 

5244.2    635.2    714.4    2949.0 

5333.0    058.2     743.       3012  5 

28 

30 

5247.2    03(5.0    715.3    2951.1 

5335.9    658.9    744."    3014.7 

30 

32 

5250.2    030.8    716.3    2953.2 

5338.8    059.7    745  "    3016.8 

32 

34 

5253.1    037.5    717.3    2955.3 

5341.8    000.5    740.       3018.9 

34 

36 

525(3.1     038.3    718.2    21)57.5 

5344.7     001.3     747.7    3021.1 

36 

38 

5259.1     639.0    719.2    2959.6 

5347.7    002.0    748.7    3023.2 

38 

40 

5262.0    639.8    720.2    2901.7 

5350.6    662.8    749.7    3025.3 

40 

4-2 

5265.0    040.6    721.1     2963.8 

5353.6    603.6    750.7    3027.5 

42 

44 

5268.0    641.3    722.1     2905.9 

5350.5    004.4    751.7    3029.0 

44 

46 

5270.9    642.1     723.1     2968.0 

5359.5    665.1     752.6    3031.7 

46 

48 

5273  9    642.8    724.1     2970.1 

5362.4    665.9    753.6    3033.8 

48 

50 

5270.9    643.6    725.0    2972.2 

5365.4    666.7    754.6    3030.0 

50 

52 

5279.8    644.4    720.0    2974.4 

5368.3    667.5    755.6    3038.1 

52 

54 

5282.8    045.1     727.0    2976.5 

5371.3    608.3    750.0    3040.2 

54 

56 

5285.8    645.9    728.0    2978.6 

5374.2    069.1     757.6    3042.4 

56 

58 

5288.7    646.6    729.0    2980.7 

5377.2    009.9    758.0    3044.5 

58 

CO 

5291.7    647.4    7299    2982.8 

5380.1     070.7    759.6    3046.6 

60 

o(i° 

57° 

L.  C.       M.         E.         T. 

L.  C.       M.         E.          T. 

/ 

0 

5380.1     670.7    759.6    3016.6 

5468.2    694.4    790.2    3111.1 

0 

5383.0    671.4     760.6    304s.  S 

5471.1     01)5.2    791.2    3113.3 

2 

4 

5:580.0    072.2     701.0     3050.9 

5474.0    096.0     792.2    3115.4 

4 

6 

5388.9    072.9    762.       3053.1 

5477.0    090.8    793.3    3117.6 

0 

8 

5391.8    073.7    763.       3055.2 

5479.9    697.6    794.3    3119.7 

8 

10 

5394.8     074.4     7G4.       3057.4 

54  82  .  8    698.4    7  95  .3    3121.  9 

10 

la 

531)7.7     075.2     705.       3059.5 

5  185.  7     099.2    796.3    3124.1 

12 

14 

5400.7    670.0    700.       3001.0 

5  KS.  7     700.0     797.4     3120.2 

14 

10 

5403.6    070.8    707.       30(33.8 

5491.6    700.8    798.4    8128.4 

16 

18 

5400.5    077.6    708.       3005.9 

5494.5    701.6    799.4    3130.6 

18 

20 

5409.5    678.4    709.       300S.1 

5497.4     702.4    800.5    3132.7 

20 

0.) 

5412.4    679.2    770.8    3070.2 

5500.3     703.  '2     801.5    3134.9 

22 

24 

5415.3    080.0     771.8    3072.4 

55J3.3    704.0    802.6    3137.0 

24 

26 

5418.3    080.8    772.8    3074.5 

5506.2    704.8    803.6    3  13!).  2 

26 

28 

5421.2    081.6    773.8    3076.6 

5501).  1     705.6    804.7    3141.4 

28 

30 

5124.1     682.4    774.8    3078.8 

5512.0     7'00.4     805.7    3143.5 

30 

aa 

5427.1     683.2    775.8    3080.9 

5515.0    707.2    800.8    3145.7 

32 

34 

5430.0    684.0    776.8    3083.1 

5517.9     708.0     S07.8    3147.9 

34 

30 

54  :i8.0    684.8    777.8    30S5.2 

5520.8     708.8     808.8    3150.0 

36 

38 

5435.9    6S5.0    778.9    3087.4 

5523.7    709.6    809.9    3152.2 

38 

40 

5438.8    080.4     779.9    3089.6 

5526.7    710.4    810.9    3154.4 

40 

42 

5441.8    087.  2     780.9    3091.7 

552!).  0     711.2    812.0    3150.6 

42 

44 

5444.7    088.0    781.9    3093.9 

5532.5    712.0    813.0    3158.7 

44 

46 

5417.6     OH8.8    783.0    3090.0 

5535.4     712.8    814.1     3100.9 

46 

48 

5450.6    689.  6    784.0    30!  IS.  2 

5538.4    713.6    815.1     3163.1 

48 

50 

5453.5    090.  4    785.0    3100.3 

5541.3    714.4    816.2    3105.3 

50 

52 

5J50.5    091.2    786.0    3102.5 

5514.2     715.2     817.2     3167.4 

52 

54 

515!).!     01)2.0     787.1     3101.  (i 

5547.1     710.0    818.3    3109.0 

54 

50 

54(52.3     U92.8     7S8.       :',10f'..8 

5550.0    716.8    819.3    3171.8 

50 

5S 

5105.3     (593.6     78'.).!     31  OS.  9 

5553.0    717.0     820.1     3174.0 

58 

00 

5408.2     094.  1     790.2     3111.1 

5555.9     718.4     821.4     317'6.1 

00_, 

IX.— 1'TXiTloXS  OF  A  ONEDEGREE  CURVE.     283 


58° 

59° 

' 

L.  C.       M.          E.           T. 

L.  C.      M.          E.          T. 

o 

5555.9    718.4    821.4    3176.1 

5643.1     742.8    8535    3241.9 

0 

2 

5558.8    719  2    822.5    3178.3 

5646.0    743.6    854.6    3244.1 

2 

4 

5561.7    720.0    823.5    3180.5 

5648.9    744.4     855.7    324C.3 

4 

6 

5564.6    720.8    824.6    3182.7 

5651.8    745.3    856.8    3248.5 

6 

8 

5567.5    721.6    825  7    3184.9 

5654.7    746.1     857.9    3250.7 

8 

10 

5570.4    722.4    826.7    3187.1 

5657.6    746.9    859.0    3252.9 

10 

1'2 

5573.3    723.2    827.8    3189.2 

5660  5    747.7    860.0    3255.1 

12 

14 

5576.2    724.0    828.9    3191.4 

5663.4    748.6    861.1     3257.3 

14 

16 

5579.2    724.8    8299    3193.6 

5666.3    749.4    862.2    3259.5 

16 

18 

5582.1     725.6    831.0    31958 

5669.2    750.2    863.3    3261.7 

18 

20 

5585.0    726.5    8321     3198.0 

5672.1     751.1     864.4    3263.9 

20 

2-2 

5587.9    727.3    833.1     3200.2 

5675.0    751.9    865-5    3266  1 

22 

24 

5590.8    728.1     834.2    3202.4 

5677.9    752.7    866.6    3268  3 

24 

26 

5MI3.7    728.9    835.3    3.  '04.  5 

5680.8    753.5    867.7    3270.5 

26 

28 

5596.6    729.7    836.3    3206.7 

5683  7    754.4    868.8    3272.7 

28 

30 

5599.5    730  5    837  4    3208.9 

5686.5    7552    869.9    3274.9 

30 

32 

5602.4    731.3    838.4    3211.1 

5689.4    756.0    871.0    3277.1 

32 

34 

5605.3    732.1     839.5    3213.3 

5692.3    756  9    872.1     3279.4 

34 

36 

5608.2    732.9    840.6    3215.5 

5695.2    757.7    873.2    3281  6 

36 

38 

5611.1     733.7    841.6    3217.7 

5698.1    758.5    874.3    3283.8 

38 

40 

5614.0    734.6    842  7    3219.9 

5701.0    759.4    875.4    3286.0 

40 

42 

5616.9    735.4    843.8    3222.1 

5703.9    760.2    876.5    3288.2 

42 

44 

5619.8    736.2    844.9    3224.3 

5700.8    761.0    877.6    3290.5 

44 

46 

5622.8    737.0    846.0    3226.5 

5709  7    761.9    878.7    3292.7 

46 

48 

5625.7    737.8    847.0    3228.7 

5712  6    762.7    879.8    3294.9 

48 

50 

5628.6    738.6    848.1    3230.9 

5715.5    763.5    880  9    3297.1 

50 

58 

5631.5    739.4    849.2    3233.1 

5718.4    764.4    882.0    3299.3 

52 

54 

5034.  4    740.2    850.3    3235.3 

5721.3    765.2    883.1     3301.5 

54 

56 

5637.3    7410    851.4    3237.5 

5724.2    766.0    884.2    3303.8 

56 

58 

5640.  2    741.9    852.5    3239.7 

5727.1     766.8    885.3    3306.0 

58 

60 

5643.1     742.8    853.5    3241.9 

5730.0    767.7    886.4    3308.2 

60 

60° 

61° 

/ 

L.  C.      M,          E.          T. 

L.  C.      M.          E.          T. 

0 

5730.0    767.7    886.4    3308.2 

5816.4    792.9    920.2    3375.2 

0 

2 

5732.9    768.5    887.5    3310.4 

5«I9.3    793.7    921.4    3377.4 

2 

4 

5735.8    709.4    888.7    3312.7 

5822.1     794.6    922.5    3379.7 

4 

6 

5738.6    770.2    889.8    3314.9 

5825.0    795.4    923.6    3381.9 

6 

8 

r>741.5    771.1     890.9    3317.1 

5827.9    796.3    924.8    3384.2 

8 

10 

5744.4    771.9    892.0    3319.3 

5830.7    797.1     925.9    S386.4 

10 

12 

5747.3    772.7    893.1     3*21.6 

5833.6    798.0    927.1     3388.7 

12 

It 

57.r>0.2    773.6    894.3    3323.8 

5836.5    798.8    928.2    3390.9 

14 

16 

5753.0    774.4    895.4    3326.0 

5839.3    799.7    929.3    3393.2 

16 

18 

5755.9    775.3    896.5    3328.3 

5842.2    800.5    930.5    3395.4 

18 

20 

5758.8    776.1     897.6    3330.5 

5845.1     801.4    931.6    3397.7 

20 

22 

5761.7    TTli.  9    898.8    3332.7 

5847.9    802.2    932.8    3399.9 

22 

24 

5764.6    777.8    899.9    3334.9 

5850.8    803.1     933.9    3402.2 

24 

26 

5767.4    778.6    901.0    3337.2 

5853.7    803.9    935.1     3404.4 

26 

28 

5770.3    779.5    902.1     3339.4 

r>85(}.5    804.8    936.3    3406.7 

28 

30 

5773.2    780.3    903.2    3341.6 

5859.4    805.6    937.4    3408.9 

30 

32 

5770.1     781.1     904.4     3343.9 

5802.3    800.5     93S.6    3411.2 

32 

34 

5779.0    782.0    905.5    3346.1 

5865.1     807.3    939.7    3413.5 

34 

36 

5781.8    782.8    906.6    334N.3 

5868.0    808.2    940.9    3415.7 

36 

38 

5784.7    783.7    907.7    3350.6 

5S70.9    809.0    942.1    3418.0 

38 

40 

5787.6    784.5    908.8    3352.8 

5873.7    809.9    943.2    3420.3 

40 

42 

57!  )0.5     785.3    910.0    3355.0 

5876.6    810.7    944.4    3422.5 

42 

44 

5793.4    786.2    911.1     3357.3 

5879.5    811.6    945.5    3424.8 

44 

46 

5796.2    787.0    912.3    3:559.5 

5S82.3    812.4    946.7    3427.1 

46 

48 

57U9.1     787.9    913.4    3361.8 

5885.2    813.3    947.8    3429.3 

48 

50 

5802.0    788.7    914.5     3364.0 

588S.1    814.1     949.0    3431.6 

50 

52 

f)S()4.9    789.5     915.7     3360.2 

iV-90.9    815.0    950.2    3433.9 

52 

54 

5807.8     7'90.4     916.8    3368.5 

5893.8    815.8    951.3     3430.1 

54 

56 

5810.6    791.2    918.0    3370.7 

riSUCi.7     816.7     U52.5    3-138.4 

56 

58 

5SI3.5     792.1     919.1     3373.0 

5K99.5    SI  7.  5     1)53.6     3440.7 

58 

60 

5816.4     792.9    920.2    3375.2 

5902.4    818.4     954.8    3442.9 

60 

284     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


62° 

63° 

/ 

L.  C.       M.        E.         T. 

L.  C.        M.         E.         T. 

0 

5902.4    818.4    954.8    3442.9 

5987.8    844.4      990.3    3511.3 

0 

2 

5905.2    819.3    956.0    3445.2 

5990.6    845.3      991.5    3513.6 

2 

4 

5908.1     820.1     957.2    3147.5 

5993.5    846.2      992.7    3515.9 

4 

6 

59109    821.0    1)583    3449.7 

BQ98.3    847.1      993.9    3518.2 

6 

8 

5913.8    821.8    959.5    3452.0 

5999.1     847.9      995.1     3520.5 

8 

10 

5916.6    822.7    960.7    3454.3 

0002  0    848.8      990.3    3522.8 

10 

12 

5919.5    823.6    961.9    3450.6 

6004.8    849.7      997.5    3525.1 

12 

14 

5922.3    824.4    903.0    3458.8 

6007.7    850.6      998.7    3527.4 

14 

16 

5925.2    825.3    904.2    3461.1 

GO  10.  5    851.4      999.9    3529.7 

16 

18 

5928.0    826.1     905.4    3463.4 

0013.3    852.3    1001.1    3532.0 

18 

20 

5930.9    8-27.0    966.6    3465.7 

0016.2    853.2    1002.3    3534.3 

20 

22 

5933.7    827.9    9G7.8    3107.9 

0019.0    854.1     1003.5    3530.0 

22 

24 

5936.6    8-28.7    908.9    3470.2 

0021.8    854.9     1004.7    3538.9 

24 

26 

5939  4    829  6    970.1     3472.5 

6024.7    855.8     1005.9    3541.2 

26 

28 

5942.3    8304    971.3    3474.7 

0027.5    856.7    1007.1    3543.5 

28 

30 

5945.1     831.3    9725    34770 

0030.3    857.6    1008.4    3545.8 

30 

32 

5947.9    832.2    973.6    3179.3 

0033.2    858.4     1009.6    3548.1 

32 

34 

5950.8    833.0'  1)7-1.8    3481.6 

0036.0    859.3    1010.8    3550.4 

34 

36 

5953.6    833.9    970.0    3483.9 

6038.9    800.2    1012.0    3552.7 

36 

38 

5956.5    834.7    977.2    3486.2 

0041.7    861.1     1013.2    3555.0 

38 

40 

5959.3    835.6    978.4    3488.5 

6044.5    861.9    1014.5    3557.3 

40 

42 

5962.2    830.5    979.6    3490.7 

6017.4    862.8     1015.7    3559.6 

42 

44 

5965.0    837.4    980.8    3493.0 

0050.2   .803.7    1016.9    3562.0 

44 

46 

5967.9    838.3    982.0    3495.3 

6053.0    864.6    1018.1     3564.3 

46 

48 

5970.7    839.1     983.2    3497.6 

0055.9    865.4     1019.3    3566.6 

48 

50 

5973.6    840.0    984.4    3499.9 

6058.7    866.3    1020.6    3508.9 

50 

52 

5976.4    810.9    985.5    3502.2 

C061.6    867.2    1021.8    3571.2 

52 

54 

5979.3    841.7    986.7    3504.5 

6004.4    868.1     1023.0    3573.5 

54 

56 

5982.1     842.6    9S7.9    3506.8 

0007.2    808.9    1024.2    3575.8 

56 

58 

5985.0    843.5    989.1     3509.0 

6070.1     869.8    1025.4    3578.1 

58 

60 

5987.8    844.4    990.3    3511.3 

6072.9    870.7    1026.7    3580.4 

60 

64°; 

65° 

L.  C.       M.          E.          T. 

L.  C.       M           E.          T. 

0 

6072.9    870.7    1026.7    3580.4 

6157.5    897.3    1064.0    3650.4 

0 

2 

6075.7    871.5     1027.9    3582.8 

6160.3    898.2    1065.2    3652.8 

2 

4 

6078.5    872.4    1029.2    &585.1 

6163.1     899.1     1066.5    3655.1 

4 

6 

6081.4    873.3    1030.4    3587.4 

6105.9    900.0    1067.7    3057.5 

6 

8 

6084.2    874.2    1031.7    3589.7 

0168.7    900.9    1069.0    3659.8 

8 

10 

6087.0    875.1     1032.9    3592.1 

6171.5    901.8    1070.2    3662.2 

10 

12 

6089.8    875.9    1034.1     3594.4 

6174.3    902.7     1071.5    3664.5 

12 

14 

609-2.6    876.8    1035.4     3596.7 

6177.1     903.6    1072.7    3666.9 

14 

16 

6095.5    877.7     1036.6    3599.1 

6179.9    91)4.5     1074.0    3669.2 

16 

18 

6098.3    878.6    1037.9    3601.4 

6182.7    905.4     1075.2    3671.6 

18 

20 

6101.1     879.5    1039.1     3603.7 

6185.5    906.3    1076.6    3673.9 

20 

22 

6103.9    880.3    1040.3    3606.0 

6188.3    907.2    1077.8    3670.2 

22 

24 

6106.7    881.2    1041.0    3608.4 

6191.1     908.1     1079.1     3078.6 

24 

26 

6109.6    882.1     1042.8    3610.7 

6193.9    909.0    1080.4    3680.9 

26 

28 

6112.4    883.0    1044.1     3013.0 

6190.7    909.9    1081.7    3683.3 

28 

30 

6115.2    883.9    1045.3    3615.3 

6199.5    910.8     1083.0    3685.6 

30 

32 

6118.0    884.7    1046.5    3617.7 

6202.3    911.7    1084.2    3688.0 

32 

34 

6120.8    885.6    1047.8    3620.0 

6-205.1     912.6     1085.5    3090.4 

34 

36 

61-23.7    886.5    1049.0    3622.3 

0-208.0    913.5    1086.8    3092.7 

36 

38 

6126.5    887.4     1050.3    3624.7 

6210.8    914.4    1088.1     301)5.1 

38 

40 

6129.3    888.3    1051.5    3627.0 

6213.6    915.3    1089.4    3697.4 

40 

42 

6132.1     889.2    1052.7    3629.4 

6216.4    916.2    1090.6    3699.8 

42 

44 

6134.9    890.1     1054.0    3631.7 

6219.2    917.1     1091.9    3702.2 

44 

46 

6137.8    891.0    1055.2    3031.0 

62-22.0    918.0    1003.2    3704.5 

46 

48 

6140.6    891.9     1050.5    3030.4 

02-24.8    918.9    1094.5    3706.9 

48 

50 

6143.4    892.8     1057.7    36:^.7 

15227.6    919.8     1095.8    3709-3 

50 

52 

6146.2    893.7     1059.0    3641.1 

(5-230.4    9-20.7    1097.0    3711.6 

52 

54 

6149.0     894.6     1000.2     3(5^.4 

1523:  J.  2    U-21.6     109R.3     3714.0 

54 

56 

6151.9    895.5     1001.5    3015.7 

(5->:J«.0    922.5    1099.6    3716.3 

56 

58 

0151.7     896.4     1002.7     364*.  1 

0238.8     1)23.4     1100.9    3718.7 

58 

60 

0157.5     897.3     1004.0    3050.4 

6241.6    924.3     1102.2    3721.1 

60 

IX.— FUNCTIONS   OF  A  ONE-DEGREE   CURVE.    285 


66° 

67° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

0 

6241.6  924.3  1102.2  3721.1 

6325.2  951.8  1141.5  :i7««.r, 

0 

2 

6244.4  925.2  1103.5  3?J3.4 

6328.0  952.7  1142.8  371)5.0 

2 

4 

6247.2  926.1  1104.8  37-5.8 

6330.7  953.6  1144.1  3797.4 

4 

6 

6250.0  927.0  1106.1  3728.  a 

6333.  5  954.5  1145.4  3799.8 

6 

8 

6252.7  927.9  1107.4  3730.6 

633(5.3  955.5  1146.7  3802.2 

8 

10 

6255.5  928.8  1108.7  3782.9 

6339.0  956.4  1148.1  3804.6 

10 

12 

6258.3  9298  1110.0  3735.3 

6341.8  957.3  1149.4  3807.0 

12 

14 

62(51.1  930.7  1111.3  3737.7 

6344.6  958.2  1150.7  3809.4 

14 

16 

6263.9  931  6  1112.6  3740.1 

6347.4  959.2  1152.0  3811.8 

16 

18 

6260.7  932.5  1113.9  3742.4 

6350.1  960.1  1153.3  3814.2 

18 

20 

6269.5  933.4  1115.2  3744.8 

6352.9  961.0  1154.7  3816.6 

20 

22 

6272.3  934.3  1116.5  3747.2 

6355.7  961.9  1156.0  3,819.0 

22 

24 

6275  0  935.3  1117.8  37-19.6 

6358.4  962.9  1157.4  3821.4 

24 

/-6 

62778  936.2  1119.1  3751.9 

6361.2  963.8  1158.7  3823.8 

26 

28 

6280.6  937.1  1120.4  3754.3 

6364.0  964.7  1160.1  38  .'6.  2 

28 

30 

6283.4  938.0  1121.7  3756.7 

6306.7  965.6  1161.4  3828.6 

30 

HB 

6286.2  938.9  1123.0  3:59.1 

6369.5  96(5.6  1162.8  3831.0 

32 

34 

62890  939.8  1124.3  3761.5 

6372.3  907.5  1164.1  3833.4 

34 

36 

6291.8  940.8  1125.6  3703.!) 

6375.1  968.4  1165.5  3835.9 

36 

38 

6294.5  941.7  1126.9  3706.3 

6377.8  909.3  1166.8  38^8.3 

38 

40 

6297.3  942  6  1128  3  3768.7 

6380.6  970.3  1168.2  3840.7 

40 

42 

6300.1  943.5  1129.6  3771.0 

0383.4  971.2  1109.5  3843.1 

42 

44 

6302.9  944.4  1130.9  3773.4 

6386.1  972.1  1170.9  3845.5 

44 

46 

6305.7  945.3  1J32.2  3775.8 

6388.9  973.0  1172.2  3847.9 

46 

48 

6308.5  946.3  1133.5  3778.2 

6391.7  974.0  1173.6  3850.4 

48 

50 

6311.3  947.2  1134.9  3780.6 

6394.4  974.9  1174.9  3852.8 

50 

52 

6314.1  948.1  1136.2  3783.0 

6397.2  975.8  1176.3  3855.2 

52 

54 

6316.8  949.0  1137.5  3785.4 

6400.0  976.8  1177.6  3857.6 

54 

56 

6319.6  949.9  1138.8  3787.8 

6402.8  977.7  1179.0  3860.0 

56 

58 

6322.4  950.8  1140.1  3790.2 

6405.5  978.6  1180.3  38625 

58 

60 

6325.2  951.8  1141.5  3792.6 

6408.3  979.6  1181.6  3864.9 

60 

68° 

69° 

/ 

L  C.   M.    E.    T. 

L.  C.    M.    E.    T. 

0 

6408.3  979.6  1181.6  3864.9 

6491.1  1007.7  1222.9  3938.1 

0 

2 

6411.1  980.5  1183.0  3867.3 

6493.8  1008.7  1224.3  3940.6 

2 

4 

6413.8  981.4  1184.4  3869.7 

6496.6  1009.6  1225.7  3943.0 

4 

6 

6416.6  982.4  1185.7  3872.2 

6499.3  1010.6  1227.1  3945.5 

6 

8 

6419.3  983.3  1187.1  3874.6 

6502.1  1011.5  1228.5  3947.9 

8 

10 

64-'2.1  984.2  1188.5  3877.0 

6504.8  1012.5  1229.9  3950.4 

10 

12 

6421.9  985.2  1189.8  3879.5 

6507.5  1013.4  1231.3  3952.9 

12 

14 

64276  986.1  1191.2  3881.9 

6510.3  1014.4  1232.7  3955.3 

14 

16 

6430.4  987.0  1192.6  3884.3 

6513.0  1015.3  1234.1  3957.8 

16 

18 

6433.1  988.0  1193.9  3886.8 

6515.8  1016.3  1235.5  3960.2 

18 

20 

6435.9  988.9  1195.3  3889.2 

6518.5  1017.2  1236.9  3962.7 

20 

22 

6438.7  989.8  1196.7  3891.6 

6521.2  1018.2  1238.3  3965.2 

22 

24 

6441.4  990.8  1198.0  3894.1 

6524.0  1019.1  1239.7  3967.6 

24 

26 

6444.2  991.7  1199.4  3896.5 

6526.7  1020.1  1241.1  3970.1 

26 

28 

6446.9  992.6  1200.8  3898.9 

6529.5  1021.0  1242.5  3972.5 

28 

30 

6449.7  993.6  1202.1  3901.4 

6532.2  1022.0  1243.9  3975.0 

30 

32 

6452.5  994.5  1203.5  3903.8 

6534.9  10-32.9  1245.3  3977.5 

32 

34 

6455.2  995.4  1204.9  3906.3 

6537.7  1023.9  1246.7  3980.0 

34 

36 

6458.0  996.4  1206.2  3908.7 

6540.4  1024.8  1248.1  3982.4 

36 

38 

6460.7  997.3  1207.6  3911.2 

6543.2  1025.8  1249.5  3984.9 

38 

40 

6463.5  998.2  1209.0  3913.6 

6545.9  1026.7  1250.9  3987.4 

40 

42 

6466.3  999.2   210.3  3916.1 

6548.6  1027.7  1252.3  3989.9 

42 

44 

6469.0  1000.1   211.7  3918.5 

6551.4  1028.6  1253.7  3092.  3 

44 

46 

6471.8  1001.0   213.1  3921.0 

6554.1  1029.6  1255.1  3994.8 

46 

48 

6474.5  1002.0   214.5  3923.4 

6556.9  1030.5  1256.5  3997  3 

48 

50 

6477.3  1002.9   215.9  3925.9 

6559.6  1031.5  1257.9  3999.8 

50 

52 

6480.1  1003.8   217.3  3928.3 

6562.3  1032.4  1259.3  4002.2 

52 

54 

6482.8  1004.8   218.7  3930.8 

0505.1  1033.4  1260.7  4004.7 

54 

56 

6485.6  1005.7   220.1  3933.2 

6567.8  1034.3  1262.1  4007.2 

56 

58 

6488.31006.7   221.5  3935.7 

6570.6  10353  1263.5  4009.7 

58 

60 

6491.1  1007.7   222.9  3938.1 

6573.3  1036.3  1005.0  4012.1 

60 

280    IX.— FUNCTIONS   OF   A  ONE-DEGREE  CURVE. 


70° 

71° 

L.  C.    M.    E.     T. 

L.  C.    M.    E     T. 

0 

6573.3  1036.3  1265.0  4012.1 

6654.9  1065.1  1308.4  4087.1 

0 

2 

6576.0  1037.3  1266.4  4014.6 

6657.6  1066.1  1309.9  4089.7 

2 

4 

6578.7  1038.2  1267.9  4017.1 

6660.3  1067.0  1311.3  4092  2 

4 

6 

6581.5  1039.2  1289.3  4019.6 

6663.0  1068.0  1312.8  4094.7 

6 

8 

6584.2  1040.1  1270.8  4022.1 

66G5.7  1068.9  1314.2  4097.2 

8 

10 

65S6.9  1041.1  1272.2  4024.6 

6668.4  1069.9  1315.7  4099.8 

10 

ft 

6589.6  1042.1  1273.6  4027.1 

6671.1  1070.9  1317.2  4102.3 

12 

14 

6592.3  1043.0  1275.1  4029.6 

6673.8  1071.9  1318.6  4104.8 

14 

16 

6595.1  1044.0  1276.5  4032.1 

6676.6  1072.9  1320.1  4107.3 

16 

18 

6597.8  1044.9  1278.0  4034.6 

6679.3  1073.8  1321.5  4109.8 

18 

20 

6600.5  1045.9  1279.4  4037.1 

6682.0  1074.8  1323.0  4112.4 

20 

22 

6603.2  1046.9  1280.8  4039.6 

6684,  T  1075.8  1324.4  4114.  'J 

22 

24 

6605.9  1047.8  1282.3  4042.1 

66874  1076.8  1325.9  4117.4 

24 

26 

6608.7  1048.8  1283.7  4044.6 

6690.1  1077.7  1327.4  4119.9 

26 

28 

6611.4  1049.7  1285.2  4047.1 

6692.8  1078.7  1328.9  4122.4 

28 

30 

6614.1  1050.7  1286.6  4049.6 

6695.5  1079.7  1330.4  4125.0 

30 

32 

6616.8  1051.7  1288.0  4052.1 

6(598.2  1080.7  1331.8  4127.5 

32 

34 

6619.5  1052.6  1289.5  4054.6 

6700.9  1081.6  1333.3  4130.4 

34 

36 

6622.3  1053.6  1290.9  4057.1 

6703.6  1082.6  1334.8  4132.6 

36 

38 

6625.0  1054.5  1292.4  4059.6 

670(5.3  1083.6  1336.3  4135.1 

38 

40 

6627.7  1055.5  1293.8  4062.1 

6709.0  1084.5  1337.8  4137.7 

40 

42 

6630.4  1056.5  1295.3  4064.6 

6711.7  1085.5  1339.2  4140.2 

42 

44 

6633.1  1057.4  1296.7  4067.1 

6714.4  1086.5  1340.7  4142.7 

44 

46 

6635.9  1058.4  1298.2.  4069.6 

6717.2  1087.5  1342.2  4145.3 

46 

48 

6638.6  1059.3  1299.6  4072.1 

6719.9  1088.4  1343.7  4147.8 

48 

50 

6641.3  1060.3  1301.1  4074.6 

6722.6  1089.4  1345.8  4150.4 

50 

52 

6644.  C  1061.3  1302.6  4077.1 

6725.3  1090.4  1346.7  4152.9 

52 

54 

6646.7  1062.2  1304.0  4079.6 

6728.0  1091.3  1348.2  4155.4 

54 

56 

6649.5  1063.2  1305.5  4082.1 

6730.7  1092.3  1349.7  4158.0 

56 

58 

6652.2  1064.1  1306.9  4084.6 

6733.4  1093.3  1351.2  4160.5 

58 

60 

6654.9  1065.1  1308.4  4087.1 

6736.1  1094.3  1352.7  4163.1 

60 

72° 

73° 

/ 

L.  C.    M.     E.    T. 

L.  C.    M.    E.    T. 

0 

6736.1  1094.3  1352.7  4163.1 

6816.6  1123.9  1398.1  4240.0 

0 

2 

6738.8  1095  2  1354.2  4165.6 

6819.3  1124.8  1399.6  4242.6 

2 

4 

6741.5  1096.2  1355.7  4168.2 

6S21.9  1125.8  1401.2  4245.1 

4 

6 

6744.1  1097.2  1357.2  4170.7 

68-J4.6  1126.8  1402.7  4247.7 

6 

8 

6746.8  1098.2  1358.7  4173.3 

6827.3  1127.8  1404.2  4250.3 

8 

10 

6749.5  1099.2  1360.2  4175.8 

6830  0  1128.8  1405.8  4252.9 

10 

12 

6752.2  1100.1  1361.7  4178.4 

6832.6  1129.8  1407.3  4255.5 

12 

14 

6754.9  1101.1  1363.2  4181.0 

6835.3  1130.8  1408.8  4258.1 

14 

16 

6757.6  1102.1  1364.7  4183.5 

6838.0  1131.8  1410.4  4260.7 

16 

18 

6760.2  1103.1  1366.2  4186.1 

6840.7  1132.8  1411.9  4263.2 

18 

20 

6762.9  1104.1  1367.7  4188.6 

6843.3  1133.8  1413.5  4265.8 

20 

22 

6765.6  1105.1  1369.2  4191.2 

6846.0  1134.8  1415.1  4268.4 

22 

24 

6768.3  1106.0  1370.7  4193.7 

6848.7  1135.8  1416.6  4271.0 

24 

26 

6771.0  1107.0  1372.2  4196.3 

6851.3  1136.8  1418.2  4273.6 

26 

28 

6773.7  1108.0  1373.7  4198.8 

6854.0  1137.8  1419.7  4276.2 

28 

30 

6776.3  1109.0  1375.2  4201.4 

6856.7  1138.8  1421.3  4278.8 

30 

32 

6779.0  1109.9  1376.7  4204.0 

6859.4  1139.8  1422.9  4281.4 

32 

34 

6781.7  1110.9  1378.2  4206.5 

68G2.0  1140.8  1424.4  4284.0 

34 

36 

6784.4  1111.9  1379.7  4209.1 

6864.7  1141.8  1426.0  4286.6 

36 

38 

6787.1  1112.9  1381.2  4211.7 

6867.4  1142.8  1427.5  4289.2 

38 

40 

6789.8  1113.9  1382.8  4214.3 

6870.1  1143.8  1429.1  4291.8 

40 

42 

6792.4  1114.9  1384.3  4216.8 

6872.7  1144.8  1430.7  4294.4 

42 

44 

6795.1  1115.9  1385.8  4219.4 

6875.4  1145.8  1432.2  4297.0 

44 

46 

6797.8  1116.9  1387.4  4222.0 

6878.1  1146.8  1433.8  4299.6 

46 

48 

6800.5  1117.9  1388.9  4224.5 

6880.8  1147.8  1435.3  4302.2 

48 

50 

6803.2  1118.9  1390.4  4227.1 

6883.4  1148  8  1436.9  4304.8 

50 

52 

6805.9  1119.9  1392.0  4229.7 

6886.1  1149.8  1438.5  4307.4 

52 

54 

IN'S.fi  1120.9  1393.5  4232.3 

6888.8  1150.8  1440.0  4310.0 

54 

56 

6811.2  1121.9  1395.0  4234.8 

6891.4  1151.8  1441.6  4312.6 

56 

58 

(5S13.9  11229  1396.6  4237.4 

6894.1  1152.8  1443.1  4315.2 

58 

60 

6816.6  1123.9  1398.1  4240.0 

6896.8  1153.8  1444.7  4317.8 

60 

IX.— FUNCTIONS  OF  A  ()XK-DK(JREE  CURVE.     287 


r~ 

74° 

75° 

L.  C.    M.    E.     T. 

L.  C.    M     E.     T. 

0 

6896.8  1153.8  1444.7  4317.8 

6976.4  1184.1  1492.5  4396.7 

0 

2 

6S99.4  1154.8  1446.2  4320.5 

6979.0  1185.1  1494.1  4399.4 

2 

4 

6902.1  1155.8  1447.8  4323.1 

6981.7  1186.1  1495.7  4402.1 

4 

6 

6904.8  1156.8  1449.4  4325.7 

6984.3  1187.1  1497.3  4404.7 

6 

8 

6907.4  1157.8  1451.0  4328.3 

6986.9  1188.1  1499.0  4407.4 

8 

10 

6910.1  1158.8  1452.6  4330.9 

6989.6  1189.2  1500.6  4410.0 

10 

12 

6912.7  1159.8  1454.1  4333.6 

6992.2  1190.2  1502.2  4412.7 

12 

14 

6915.4  1160.8  1455.7  4336.2 

6994.9  1191.2  1503.8  4415.3 

14 

16 

6918.0  1161.8  1457.3  4338.8 

6997.5  1192.2  1505.4  4418.0 

16 

1.8 

6920.7  1162.8  1458.9  4341.4 

7000.1  1193.2  1507.0  4420.7 

18 

20 

6923.3  1163.9  1460.5  4344.0 

7002.8  1194.3  1508.7  4423.3 

20 

22 

6926.0  1164.9  1462.0  4346.7 

7005.4  1195.3  1510.3  4426.0 

22 

24 

6928.6  1165.9  1463.6  43-19.3 

7008.0  1196.3  1512.0  4428.6 

24 

26 

6931.3  1166.9  1465.2  4351.9 

7010.7  1197.3  1513.6  4431.3 

26 

28 

6933.9  1167.9  1466.8  4354.5 

7013.3  1198.3  1515.3  4434.0 

28 

30 

6936.6  1168.9  1468.4  4357.1 

7015.9  1199.4  1516.9  4436.6 

30 

32 

6939.2  1169.9  1469.9  4359.8 

7018.6  1200.4  1518.5  4439.3 

32 

34 

6941.9  1170.9  1471.5  4362.4 

7021.2  1201.4  1520.2  4442.0 

34 

36 

6944.6  1171.9  1473.1  4365.1 

7023.9  1202.4  1521.8  4444.6 

36 

38 

6947.2  1172.9  1474.7  4367.7 

7026.5  1203.4  1523.5  4447.3 

38 

40 

6949.9  1174.0  1476.4  4370.3 

7029.1  1204.5  1525.1  4450.0 

40 

42 

6952.5  1175.0  1478.0  4373.0 

7031.8  1205.5  1526.7  4452.7 

42 

44 

6955.2  1176.0  1479.6  4375.6 

7034.4  1206.5  1528.4  4455.3 

44 

46 

6957.8  1177.0  1481.2  4378.3 

7037.0  1207.5  1530.0  4458.0 

46 

48 

6960.5  1178.0  1482.8  4380.9 

7039.7  1208.5  1531.7  4460.7 

48 

50 

6963.1  1179.0  1484.4  4383.5 

7042.3  1209.6  1533.3  4463.4 

50 

52 

6965.8  1180.0  1486.0  4386.2 

7045.0  1210.6  1534.9  4466.0 

52 

54 

6968.4  1181.0  1487.7  4388.8 

7047.6  1211.6  1536.6  4468.7 

54 

56 

6971.1  1182.0  1489.3  4391.5 

7050.2  1212.6  1538.2  4471.4 

56 

58 

6973.7  1183.0  1490.9  4394.1 

7052.9  1213.6  1539.9  4474.1 

58 

60 

6976.4  1184.1  1492.5  4396.7 

7055.5  1214.7  1541.5  4476.7 

60 

76° 

77° 

L.  C.    M.    E.     T. 

L.  C.    M     E.     T. 

0 

7055.5  1214.7  1541.5  4476.7 

7134.0  1245.6  1591.7  4557.8 

0 

2 

7058.1  1215.7  1543.2  4479.4 

7136.6  1246.6  1593.4  4560.5 

2 

4 

7060.7  1216.7  1544.9  4482.1 

7139.2  1247.7  1595.1  4563.3 

4 

6 

7063.3  1217.8  1516.5  4484.8 

7141.8  1248.7  1596.8  4506.0 

6 

8 

7006.0  1218.8  1548.2  4487.5 

7144.4  1249.8  1598.5  4568.7 

8 

10 

7008.6  1219.8  1549.9  4490.2 

7147.0  1250.8  1600.2  4571.5 

10 

12 

7071.2  1220.9  1551.5  4492.9 

7149.6  1251.8  1601.9  4574.2 

12 

14 

7073.8  1221.9  1553.2  4495.6 

7152.2  1252.9  1603.6  4576.9 

14 

16 

7076.4  1222.9  155J.9  4498.3 

7154.8  1253  9  1605.3  4579.7 

16 

18 

7079.0  1224.0  1556.5  4501.0 

7157.4  1255  0  1607.0  4582.4 

18 

20 

7081.7  1225.0  1558.2  4503.7 

7160.0  1256.0  1608.7  4585.1 

20 

22 

7084.3  1226.0  1559.9  4500.  S 

7162.6  1257.0  1610.4  4587.9 

22 

24 

7086.9  1227.1  1561.5  4509.0 

7165.2  1258.1  1612.1  4<50  6 

24 

20 

7089.5  1228.1  1563.2  4511.7 

7167.8  1259.1  1613.8  4593.3 

26 

28 

7092.1  1229.1  1564.9  4514.4 

7170.4  1260.2  1615.5  4596.0 

28 

30 

7094.7  1230.2  1566.5  4517.1 

7173.0  1261.2  1617.3  45988 

30 

32 

7097.4  1231.2  1568.2  4519.8 

7175.6  1262.2  1619.0  4601.5 

32 

31 

7100.0  1232.2  1569.9  4522.5 

71782  1263.3  1620.7  4604.3 

31 

36 

7102.6  1233.3  1571.5  4525.3 

7180.8  1264.3  1622.4  4607.0 

'36 

38 

7105.2  1234.3  1573.2  4528.0 

7183.4  1265.4  1624.1  4609.8 

38 

40 

7107.8  1235.3  1574.8  4530.7 

7186.0  1266.4  1625.9  4612.5 

40 

42 

7110.4  1236.4  1576.4  4533.4 

7188.6  1267.4  1627.6  4615.3 

42 

44 

7113.1  1237.4  1578.1  4536.1 

7191.2  1268.5  1629.3  4618.0 

44 

46 

7115.7  12384  1579.8  4538.8 

7193.8  1269.5  1631.0  4620.8 

46 

48 

7118.3  1239  5  1581.5  4541.5 

7196.4  12706  1632.7  4623.5 

48 

50 

7120.9  1240  5  1583.2  4544.2 

7199.0  1271.6  1634  5  4626.3 

50 

B2 

7123.5  1241.5  1584.9  4547.0 

7201.6  1272.7  1636.2  4029.0 

52 

54 

7126.1  1242.6  1536.6  4549.7 

7204  2  1273.7  1637.9  4631.8 

54 

56 

7128.8  1243.6  1588.3  4552.4 

7206.  S  1274.8  1639.6  4634.5 

56 

58 

7131.4  1244.6  15900  4555.1 

7209.4  1275.8  1641.3  4637.3 

58 

60 

7134  0  1-.M5.6  1591.7  4557.8 

7212.0  1276.9  1643.1  4640.0 

60 

OF  THB 


288     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


78° 

79° 

L.  C.   M.    E.    T. 

L.  C.    M.     E.    T. 

0 

7212.0  1276.9  1643.1  4»;»u.O 

7289.5  1308.5  1696.0  4723.  1 

0 

o 

7214.6  1278.0  1644.8  4642.8 

7292.1  1309.5  1697.7  4726.2 

4 

7217.2  1279.0  1640.  G  4045.6 

72JH.G  1310.6  1099.5  4729.0 

4 

6 

7219.7  1280.1  1648.3  4648.3 

72J»7.2  1311.7  1701.3  4731.8 

0 

8 

7222.3  1281.1  1650.1  4651.1 

7299.7  1312.7  1703.1  4734.7 

8 

10 

7224.9  1282.2  1651.8  4653.9 

7302.3  1313.8  1704.9  4737.5 

10 

IS 

7227.5  1283.2  1653.6  4656.7 

7304.9  1314.9  17'06.6  4740.3 

12 

14 

7230.1  1284.3  1655.3  4659.4 

7307.4  1315.9  1708.4  4743.1 

14 

16 

7232.7  1285.3  1057.1  4662.2 

7310.0  1317.0  1710.2  4745.9 

16 

18 

7235.2  1286.4  1658.8  4665.0 

7312.6  1318.1  1712.0  4748.7 

18 

20 

7237.8  1287.4  1660.6  4667.7 

7315.1  1319.1  1713.8  4751.5 

20 

22 

7240.4  1288.5  1662.3  4670.5 

7317.7  1320.2  1715.6  4751.3 

22 

24 

7243.0  1289.5  1664.1  4673.3 

7320.3  1321.3  1717.4  4757.1 

24 

26 

7245.6  1290.6  1665.8  4676.0 

7322.8  1322.3  1719.2  4700.0 

26 

28 

7248.2  1291.6  1667.6  4678.8 

7325.4  1323.4  1721.0  4762.8 

28 

30 

7250.7  1292.7  1669.3  4681.6 

7327.9  1324.5  1722.8  4765.6 

30 

32 

7253.3  1293.7  1671.1  468-J.4 

7330.5  1325.5  1724.6  4768.4 

32 

34 

T255.9  1294.8  1672.8  4687.2 

7333.1  1326.6  1720.   4771.2 

34 

36 

7258.5  1295.8  1674.6  4089.9 

7335.6  1327.7  1728.2  4774.1 

36 

38 

7261.1  1296.9  1676.3  4692.7 

7338.2  1328.7  1730.0  4776.9 

38 

40 

7263.7  1297.9  1678.2  4695.5 

7340.8  1329.8  1731.9  4779.7 

40 

42 

7266.2  1299.0  1679.9  4698.3 

7343.3  1330.8  1733.7  4782.6 

4^< 

44 

7268.8  1300.0  1681.7  4701.1 

7345.9  1331.9  1735.5  4785.4 

44 

46 

7271.4  1301.1  1683.5  4703.9 

7348.4  1333.0  1737.3  4788.2 

46 

48 

7274.0  1302.1  1685.3  4706.7 

7351.0  1334.1  1739.1  4791.0 

48 

50 

7276.6  1303.2  1687.1  4709.5 

7353.6  1335.2  1740.9  4793.9 

50 

52 

7279.2  1304.2  1688.8  4712.2 

7350.1  1336.2  1742.7  4796.7 

52 

54 

7281.7  1305.3  1690.6  4715.0 

7358.7  1337.3  1744.5  4799.5 

54 

56 

7284.3  1306.3  1692.4  4717.8 

7301.3  1338.4  1746.3  4802.4 

56 

58 

7286.9  1307.4  1694.2  4720.6 

7363.8  1339.5  1748.1  4805.2 

58 

60 

7289.5  1308.5  1696.0  4723.4 

7366.4  1340.6  1750.0  4808.0 

60 

80° 

81° 

L.  C.   M.     E.     T. 

L.  C.   M.     E.     T. 

0 

7366.4  1340.6  17500  4808.0 

7442.7  1372.8  1805.5  4893.9 

0 

2 

7368  9  1341.7  1751.8  4810.9 

7445.2  1373.9  1807.3  4896.8 

2 

4 

7371.5  1342.7  1753.7  4813.7 

7447.7  1375.0  1809.2  4899.7 

4 

6 

7374.0  1343.8  1755.5  4816.6 

7450.3  1376.1  1811.1  4902.6 

6 

8 

7376.6  1344.9  1757.4  4819.4 

7452.8  1377.1  1813.0  4905.4 

8 

10 

7379.1  1346.0  1759.2  4822.3 

7455.3  1378.2  1814.9  4908.3 

10 

12 

7381.7  1347.0  1761.0  4825.1 

7457.8  1379.3  1816.8  4911.2 

12 

14 

7384.2  1348.1  1762.9  4828.0 

7460.4  1380.4  1818.6  4914.1 

14 

16 

7386.7  1349.2  1764.7  4830.8 

7462.9  1381.4  1820.5  4917.0 

16 

18 

7389.3  1350.3  1766.6  4833.7 

7465.4  1382.5  1822.4  4919.9 

18 

20 

7391.8  1351.3  1768.4  4836.5 

7467.9  1383.6  1824.2  4922.8 

20 

22 

7394.4  1352.4  1770.2  4839.4 

7470.4  1384.7  1826.1  4925.7 

22 

24 

7396.9  1353.5  1772.1  4842.2 

7473.0  1385.7  1828.0  4928.6 

24 

26 

7399.5  1354.6  1773.9  4845.1 

7475.5  1386.8  1829.9  4931.5 

26 

28 

7402.0  1355.6  1775.8  4847.9 

7478.0  1387.9  1831.8  4934.4 

28 

30 

7404.5  1356.7  1777.6  4850.8 

7480.5  1389.0  1833.7  4937.2 

30 

32 

7407.1  1357.8  1779.4  4853.7 

7483.1  1390.1  1835.6  4940.2 

32 

34 

7409.6  1358.9  1781.3  4856.5 

7485.6  1391.2  1837.5  4943.1 

34 

36 

7412.2  1359.9  1783.1  4859.4 

7488.1  1392.3  1839.4  4946.0 

36 

38 

7414.7  1361.0  1785.0  4862.3 

7490.6  1393.4  1841.3  4948.9 

38 

40 

7417.3  1362.1  1786.8  4865.1 

7493.2  1394.5  1843.2  4951.8 

40 

42 

7419  8  1363.2  1788.6  4868.0 

7495.7  1395.6  1845.1  4954.7 

42 

44 

7422.3  1364.2  1790.5  4870.9 

7498.2  1396.7  1847.0  4957.6 

44 

46 

7424.9  1365.3  1792  4  4873.8 

7500.7  1397.8  1848.9  4960.6 

46 

48 

7427.4  1366.4  1794  3  4876.6 

7503.3  1398.9  1850.8  4963.5 

48 

50 

7430.0  1367.5  1796.2  4879.5 

7505.8  1400.0  1852  7  4966.4 

50 

52 

7432.5  1368.5  1798.0  4882.4 

7508.3  1401.1  1854.6  4969.3 

52 

54 

7435.1  1369.6  1799.9  4885.3 

7510.8  1402.2  1856.5  4972.2 

54 

56 

7437.6  1370  7  1801.8  4888.1 

7513.3  1403.3  1858.4  4975.1 

56 

58 

7440.1  1371.?  1803.7  4891.0 

7515.9  1404  4  1860.3  4978.0 

58 

60 

7442.7  1372.8  1805.5  4893.9 

7518.4  1405.5  1862  3  4981.0 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.      289 


S-2° 

83° 

/ 

L.  0.    M.     E.     T 

L.  C.   M.     E.     T. 

0 

7518.4  1405.5  1862.3  49S1.0 

7593.6  1438.5  1920.6  5069.4 

0 

2 

7520.9  1406.6  1864.2  49S3.9 

7596.1  1439.6  1922.6  5072.4 

2 

4 

7523.4  1407.7  1866.1  498(5.8 

7598.6  1440.7  1924.6  5075.4 

4 

6 

7525.9  1408  8  1868.1  4989.8 

7601.1  1441.8  1926.5  5078.4 

6 

8 

7528.4  1409.9  1870.0  4992.7 

7603.6  1442.9  1928.5  5081.4 

8 

10 

7530.9  1411.0  1871.9  4995.7 

7606.0  1444.0  1930.5  5084.4 

10 

12 

7533  4  1412.1  1873  9  4998.6 

7608.5  1445.1  1932.4  5087.3 

12 

14 

7535.9  1413.2  1875.8  5001.5 

7611.0  1446.2  1934.4  5090.3 

14 

16 

7538.5  1414.3  1877.7  5004.5 

7613.5  1447.3  1936.4  5093.3 

16 

18 

7541.0  1415.4  1879.7  5007.4 

7616.0  1448.4  1938.4  5096.3 

18 

20 

7543.5  1416.5  1881.6  5010.3 

7618.5  1449.6  1940.4  5099.3 

20 

22 

7546.0  1417.6  1883.5  5013.3 

7621.0  1450.7  1942.4  5102.3 

22 

24 

7548.5  1418.7  1885.5  5016.2 

7623.5  1451.8  1944.4  5105.2 

24 

26 

7551.0  1419.8  1887.4  5019.2 

7626.0  1452.9  1946.4  5108.2 

26 

28 

7553.5  1420.9  1889.3  502'.'.  1 

7628.5  1454.0  1948.4  5111.2 

28 

30 

7556.0  1422.0  1891.3  5025.0 

7630.9  1455.1  1950.4  5114  2 

30 

32 

7558.5  1423.1  1893.2  5028.0 

7633.4  1456.2  1952.4  5117.2 

32 

34 

7561.0  1424.2  1895.1  5031.0 

7635.9  1457.3  1954.4  5120.2 

34 

36 

7563.5  1425.3  1897.1  5033.9 

7638.4  1458.4  1956.4  5123.2 

36 

38 

7566.0  1426.4  1899.0  5036.9 

7640.9  1459.5  1958.4  5126.2 

38 

40 

7568.5  1427.5  1901.0  5039.8 

7643.4  1460.7  1960.4  5129.2 

40 

42 

7571.0  1428.6  1902.9  504','.  8 

7645.9  1461.8  1962.   5132.2 

42 

44 

7573.5  1429.7  1901.9  5045.8 

7648.4  1462.9  1964.   5135.2 

44 

46 

7576.1  1430.8  1906.9  5048.7 

7650.9  1464.0  1966.   5138.2 

46 

48 

7578.6  1431.9  1908.8  5051.7 

7653.4  1465.1  1968.   5141.2 

48 

50 

7581.1  1433.0  1910.8  5054.6 

7655.8  1466.2  1970.   5144.3 

50 

52 

7583.6  1434.1  1912.8  5057.6 

7658.3  1467.3  1972.   5147.3 

52 

54 

7586.1  1435.2  1914.7  5060.6 

7660.8  1468.4  1974.   5150.3 

54 

56 

7588.6  1436.3  1916.7  5063.5 

7663.3  1469.5  1976.4  5153.3 

56 

58 

7591.1  1437.4  1918.7  5066.5 

7665.8  1470.6  1978.4  5156.3 

58 

60 

7593.6  1438.5  1920.6  5069.4 

7668.3  1471.8  1980.5  5159.3 

60 

84° 

85° 

' 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     1 

0 

7668.3  1471.8  1980.5  5159.3 

7742.4  1505.4  2041.8  5250.6 

0 

2 

7670.8  1472.9  1982.5  5162.3 

7744.8  1506.5  2043.9  5253.6 

2 

4 

7673.2  1474.0  1984.5  5165.3 

7747.3  1507.6  2046.0  5256.7 

4 

6 

7675.7  1475.1  1986.6  5168.4 

7749.7  1508.8  2048.0  5259.8 

6 

8 

7678.2  1476.2  1988.6  5171.4 

7752.2  1509.9  2050.1  5262.9 

8 

10 

7680.6  1477.4  1990.6  5174.4 

7754.6  1511.0  2052.2  5266  0 

10 

12 

7683.1  1478.5  1992.7  5177.5 

7757.1  1512.2  2054.2  5269.0 

12 

14 

7685.6  1479.6  1994.7  5180.5 

7759  5  1513.3  2056.3  5272.1 

14 

16 

7688.1  1480.7  1996.7  5183.5 

7762.0  1514.4  2058.4  5275.2 

16 

18 

7690.5  1481.8  1998.8  5186.6 

7764  4  1515.6  2060.5  5278.3 

18 

20 

7693.0  14&3.0  2000.8  5189.6 

7766.9  1516.7  2062.6  5281.4 

20 

22 

7695.5  1484.1  2002.8  5192.6 

7769.3  1517.8  2064.7  5284.4 

22 

24 

7697.9  1485.2  2004.9  5195.6 

7771.8  1519.0  2066.8  5287.5 

24 

26 

7700.4  1486.3  2006.9  5198.7 

7774.2  1520.1  2068.9  5290.6 

26 

28 

7702.9  14874  2008.9  5201.7 

7776.7  1521.2  2071.0  5293.7 

28 

30 

7705.3  14886  2011.0  5204.7 

7779.1  1522.4  2073.1  5296.7 

30 

32 

7707  8  1489.7  2013.0  5207.8 

7781.5  1523.5  2075.2  5299.8 

32 

34 

7710.3  1490.8  2015.0  5210.8 

7784.0  1524.6  2077  3  5302.9 

34 

36 

7712.8  1491.9  2017.0  5213.9 

7786.4  1525.8  2079.4  5306.1 

36 

38 

7715.2  1493.0  2019.1  5216.9 

7788.9  1526.9  2081.5  5309.2 

38 

40 

7717.7  1494.2  2021.2  5220.0 

7791.3  1528.0  2083.7  5312.3 

40 

42 

7720.2  1495.3  2023.2  5223.1 

7793.8  1529.2  2085.8  5315.4 

42 

44 

7722.6  1496.4  2025.3  5226.1 

7796.2  1530.3  2087.9  5318.5 

44 

46 

7725.1  1497.5  2027.4  5229.2 

77987  1531.4  2090.0  5321.6 

46 

48 

7727.6  1498.6  2029.4  5232.2 

7801.1  1532.6  2092.1  5324.7 

48 

50 

7730.0  14998  2031.5  5235.3 

7803.6  1533.7  2094.2  5327.8 

50 

52 

7732  5  1500.9  2033.6  5238.3 

7806.0  1534  8  2096.3  5330  9 

52 

54 

7735.0  1502.0  2035.6  5241  4 

7808.5  1536.0  2098.4  5334.0 

54 

56 

7737.5  1503.1  2037.7  5244  5 

7810.9  1537.1  2100.6  5337.1 

56 

58 

7739  9  1504.2  2039.8  5247.5 

7813  4  1538.2  2102.7  5340.2 

58 

60 

7742.4  15054  2041.8  5250.6 

7815  8  li.39.3  2104.8  5343.3 

60 

290      IX.— FUNCTIONS  OF  A  ONE-PE<JKEE  CURVE. 


86° 

87° 

L.  C.        M.          E.          T. 

L.  C.       .M.          E.          T. 

0 

7815.8    1539.3    2104.8    5343.3 

7888.5    1573.6    2169.5    5437.5 

0 

2 

7818.2     1540.4    2106.9    5346.4 

7890.9    1574.8    2171.6    5440.7 

2 

4 

7820.6    1541.6    2109.1    5349.5 

7893.3    1575.9    2173.8    5443.!) 

4 

6 

7823.1     1542.7    2111.2    5352.7 

7895.7    1577.1     2176.0    5447.1 

6 

8 

7825.5     1543.9    2113.4    5355.8 

7898.1     1578.2    2118.2    5450.3 

8 

10 

7827.9    1545.0    2115.5    5358.  9 

7900.5    1579.4    2180.4    5453.4 

10 

12 

7830.3    1546.1     2117.6    5362.0 

7903.0    1580.5    2182.5    5456.6 

12 

14 

7832.8    1547.3    2119.8    53'J5.2 

7905.4     1581.7    2184.7    5459.8 

14 

16 

7835.2    1548.4    2121.9    53(58.3 

7907.8     1582.9    2186.9    5463.0 

16 

18 

7837.6    1549.6    2124.1     5371.4 

7910.2    1584.0    2189.1     5466.2 

18 

20 

7840.0    1550.7    2126.2    5374.6 

7912.6    1585.1     2191.3    5469.4 

20 

22 

7842.4    1551.8    2128.3    5377.7 

7915.0    1586.3    2193.5    5472.5 

22 

24 

7844.9     1553.0    2130.5    5  80.8 

7917.4    1587.4    2195.7    5475.7 

24 

26 

7847.3     1554.1     2132.6    5383.9 

7919.8    1588.6    2197.9    5478.9 

26 

28 

7849.7    1555.3    2134.8    5387.1 

7922.2    1589.7    2200.1     5482.1 

28 

30 

7852.1     1556.4    2136.9    5390.2 

7924.6    1590.9    2202.3    5485.3 

30 

32 

7854.6    1557.5    2139.0    5393.4 

7927.1     1592.0    2204.5    5488.5 

32 

34 

7857.0     1558.7    2141.2    539G.5 

7929.5     1593.2    2208.8    5491.7 

34 

36 

7859.4    1559.8    2143.3    53'.)'.).  7 

7931.9    1594.3    2209.0    5494.9 

36 

38 

7861.8     1561.0    2145.5    5402.8 

7934.3     1595.5    2211.2    5498.1 

38 

40 

7864.3    1562.1     2147.7    5406.0 

7936.7    1596.6    2213.4    5501.3 

40 

42 

7866.7    1563.2    2149.8    5409.1 

7939.1     1597.8    2215.6    5504.5 

42 

44 

7869.1     1564.4    2152.0    5412.3 

7941.5    1598.9    2217.8    5507.7 

44 

46 

7871.5     1565.5    2154.2    5415.4 

7913.9     1600.1     2220.0    5510.9 

46 

48 

7874.0    1566.7    2156.4    5418.6 

7946.3    1601.2    2222.3    5514.1 

48 

50 

7876.4    1567.8    2158.6    5121.8 

7948.7    1602.4    2'.'24.5    5517.3 

50 

52 

7878.8    1568.9    2160.7    5424.9 

7951.2    1603.5    2226.7    5520.5 

52 

54 

7881.2    1570.1     2162.9    5428.1 

7953.6     1604.7    2228.9    5523.7 

54 

56 

7883.6     1571.2    2165.1     5431.2 

7956.0     1605.8    2231.1     5526.9 

56 

58 

7886.1     1572.4    2107.3    5134.4 

7958.4     1607.0    2233.3    5530.1 

58 

60 

788S.5    1573.6    2169.5    5137.5 

7960.8    1608.2    2235.6    5533.3 

60 

88° 

89° 

/ 

L.  C.        M.          E.           T. 

L.  C.        M.          E.           T. 

0 

7960.8    1608.2    2235.6    5533.3 

8032.4    1643.0    2303.6    5630.8 

0 

2 

7963.2    1609.4    2237.8    5536.6 

8034.8    1644.1    2305.9    5631.1 

2 

4 

7965.6     1610.5    2240.1     5539.8 

8037.1     1645.3    2308.2    5637.4 

4 

6 

7968.0    1611.7    2J42.3    5543.1 

8039.5    1646.5    2310.5    5640.7 

6 

8 

7970.3    1612.8    2'344.6    5546.3 

8041.9     1647.7    2312.8    5644.0 

8 

10 

7972.7    1614.0    2246.8    5549.5 

8044.2    1648.9    2315.1    5647.3 

10 

1? 

7975.1     1615.2    2249.1     5552.8 

8046.6    1650.0    2317.4    5650.6 

12 

14 

7977.5    1616.3    2251.3    5556.0 

8049.0    1651.2    2319.7    5653.9 

14 

16 

7979.9     1617.5    2253.6    5559.2 

8051.4    1652.4    2322.0    5657.1 

16 

IS 

7982.3    1618.6    2255.8    5562.5 

8053.7    1653.6    2324.3    5660.4 

18 

20 

7984.7    1619.8    2258.1     5565.7 

8056.1     1654.8    2326.7     5663.7 

20 

22 

7987.1     1621.0    2260.4    5568.9 

8058.5    1655.9    2329.0    5667.0 

22 

24 

7989.4    1622.1     2262.7    5572.2 

8060.8     1657.1    2331.3    5670.3 

24 

26 

7991.8     1623.3    2264.9    5575.4 

8063.2    1658.3    2333.7    5673.6 

26 

28 

7994.2    1624.4    2267.2    5578.6 

8065.6    1659.5    2336.0    5676.9 

28 

30 

7996.6    1625.6    2269.5    5581.9 

8067.9    1660.7    2338.3    5680.2 

30 

32 

79W.O    1626.8    2271.7    5585.1 

8070.3    1661.8    2340.7    5683.5 

32 

34 

8001.4     1627.9    2273.9    5588.4 

8073.7    1663.0    2343.0    5686.8 

34 

36 

8003.8     1629.1     2276.2    5591.7 

8075.1     1664.2    2345.3    5690.2 

36 

38 

8006.1     1630.2    2278.5    5594.9 

8077.4    1665.4    2347.7    5693.5 

38 

40 

8008.5    1631.4    2280.8    5598.2 

8079.8    1666.6    2350.0    5696.8 

40 

42 

8010.9     1632.6    2283.0    5601.4 

8082.  2    1667.7    2352.3    5700.1 

42 

44 

8013.3    1633.7    2285.3    5604.7 

8084.5    1668.8    2354.7    5703.4 

44 

46 

8015.7    16:34.9    2C'87.6    5608.0 

8086.9    1670.0    2357.0    5106.8 

46 

48 

8018.1     1636.0    2289.9    5611.2 

8089.3    1671.2    2359.3    5710.1 

48 

50 

8020.5    1637.2    2292.2    5614.5 

8091.6    1672.4    2361.7    5713.4 

50 

52 

8022.9    1638.4    2294.4    5617.8 

8094.0    1673.5    2364.0    5716.7 

52 

54 

8025  .  2     1  639  .  5    2296  .  7     562  1  .  0 

8096.4     1674  7    2366.3    5720.0 

54 

56 

8027.6    1640.7    22SW.O    5621.3 

8098.8    1675.9    2368.7    5723.  4 

56 

58 

8030.0    1641.8    2301.3    5(K'7.5 

8101.1     1677.1     2371.0    5726.7 

18 

60 

8032.4     1643.0    2303.6    5630.  S 

8103.5     1078.3    2373.4     57300 

no 

IX.— FUNCTIONS  OF  A  ONE  DWJRKK  CURVE.      291 


10° 

ill  ' 

' 

L.  C.    M.     E.     T. 

L.  C.   M.    E.     T. 

0 

8103.5  1678.3  2373.4  5730.0 

8173.9  171^.8  2445.1  5S30.9 

0 

8105.8  1679.5  2375.8  5733.3 

8176.2  1715.0  2447.5  5834.3 

0 

4 

8108.2  16S0.6  2378.2  5736.7 

8178.5  1716.2  2450.0  5837.7 

4 

6 

8110.5  1081.8  2380.5  5740.0 

8180.9  1717.4  2452.4  5841.1 

6 

8 

8112.9  1683.0  2382.9  5743.4 

8183.2  1718.6  2454.8  5844.5 

8 

10 

8115.2  1684.2  2385.3  5746.7 

81S5.5  1719.7  2457.2  5847.9 

10 

8117.6  1685.4  2387.6  5750.0 

8187.9  1720.9  2459.7  5851.3 

12 

14 

8119.9  1C86.5  2390.0  5753.4 

8190.2  1722.1  2402.1  5854.7 

14 

16 

8122.3  1687.7  2392.4  5756.7 

8192.5  1723.3  2464.5  5858.1 

16 

18 

8124.6  1688.9  2394.7  5760.1 

8194.8  1724.5  2467.0  5861.5 

18 

20 

8127.0  1690.1  2397.1  5763.4 

8197.2  17-25.7  2469.4  5864.9 

20 

22 

8129.3  1691.3  2399.5  576(5.8 

8199.5  1726.9  2471.9  5868.3 

22 

•-.'4 

8131.7  1692.5  2401.9  5770.1 

8201.8  1728.1  2474.3  5871.8 

24 

26 

8134.0  1693.6  2404.3  5773.5 

8204.2  1729.3  2476.7  5875.2 

26 

28 

8136.4  1094.8  2406.6  5776.9 

8206.5  1730.5  2479.2  5878.6 

28 

30 

8138.7  1696.0  2409.0  5780.2 

8208.8  1731.7  2481.6  5882.0 

30 

32 

8141.1  1697.2  2411.4  5783.6 

8211.1  1732.9  2484.1  5885.4 

32 

34 

8143.4  1698.4  2413.8  5787.0 

8213.5  1734.1  2486.5  5888.9 

34 

36 

8145.8  1099.6  2416.2  5790.3 

8215.8  1735.3  2489.0  5892.3 

36 

38 

8148.1  1700.7  2418.6  5793.7 

8218.1  1736.4  2491.5  5895.7 

38 

40 

8150.4  1701.9  2421.0  5797.1 

8220.4  1737.6  2493.9  5899.2 

40 

4:2 

8152.8  1703.1  2423.4  5800.4 

8222.8  1738.8  2496.4  5902.6 

42 

44 

8155.1  1704.3  2425.8  5803.8 

8225.1  1740.0  2498.9  5906.0 

44 

46 

8157.5  1705.5  2428.2  5807.2 

8227.4  1741.2  2501.3  5909.4 

46 

48 

8159.8  1706.7  2430.6  5810.6 

8229.7  1742.4  2503.8  5912.9 

48 

50 

8162.2  1707.9  2433.0  5814.0 

8232.0  1743.6  2506.3  5916.3 

50 

52 

8164.5  1709.0  2435.4  5817.3 

8234.3  1744.8  2508.7  5919.8 

52 

54 

8166.8  1710.2  2437.9  5820.7 

8230.7  1740.0  2511.2  5923.2 

54 

56 

8169.2  1711.4  2440.3  5824.1 

8239.0  1747.2  2513.7  5926.7 

56 

58 

8171.5  1712.6  2442.7  5827.5 

8241.3  1748.4  2516.2  5930.1 

58 

60 

8173.9  1713.8  2445.1  5830.9 

8-243.6  1749.6  2518.7  5933.6 

60 

92° 

93° 

L.  C    M.    E.     T. 

L.  C.   M.    E.     T. 

0 

8243.6  1749.6  2518.7  5933.6 

8312.8  1785.7  2594.2  6038.2 

0 

2 

8245.9  1750.8  2521.2  5937.0 

8315.1  1786.9  2596.8  6041  7 

2 

4 

8248.2  1752.0  2523.6  5940.5 

8317.4  1788.2  2599.3  6045.2 

4 

6 

8250.6  1753.2  2526.1  5944.0 

8319.7  1789.4  2601.9  6048.7 

6 

8252.9   754.4  2528.6  5947.4 

8322.0  1790.0  2004.4  6052  2 

8 

10 

8255.2   755.6  2531.1  5950.9 

8324.3  1791.8  2007.0  6055.8 

10 

12 

8257.5   756.8  2533.6  5954.4 

83-26.6  1793.0  2609.6  6059.3 

12 

14 

8259.8   758.0  2536.1  5957.8 

8328.8  1794.2  2612.1  6062.8 

14 

16 

8262.2   759.2  2538.6  5961.3 

8331.1  1795.4  2614.7  6060.4 

16 

18 

8264.5   760.4  2541.1  5964.8 

8333.3  1796.6  2617.3  6009.9 

18 

20 

8266.8   761.6  2543.6  5968.2 

8335.6  1797.8  2619.8  6073.4 

20 

22 

8269.1   702.8  2546.1  5971.7 

8337  9  1799.1  2622.4  6077.0 

22 

24 

8271.4   764.0  2548.6  5975.2 

8340.2  1800.3  2625.0  6080.5 

24 

26 

8273.7   765.2  2551.  2  5978.7 

8342.5  1801.5  2627.6  6084.1 

26 

28 

8276.0   700.4  2553.7  5982.2 

8341.8  1802.7  2030.2  6087.6 

28 

30 

8278.3   767.6  2556.2  5985.6 

8347.1  1803.9  2632.7  6091.2 

30 

32 

8280.6   7  OS.  8  2558.7  5989.1 

8319.4  1805.1  2035.3  6094.7 

32 

34 

8282.9   7;0.0  2501.2  5992.6 

8  ;r>l.  7  1806.3  2037.9  6098.3 

34 

30 

8285.2   771.2  2563.8  5990.1 

8354.0  1807.6  2640.5  6101.8 

36 

38 

8287.5   772.5  2506.3  5999.6 

8356.3  1808.8  2643.1  6105.4 

38 

40 

8289.8   773.7  2568.8  6003.1 

8358.5  1810.0  2645.7  6109.0 

40 

42 

8292.1   774.9  2571.3  0000.6 

8300.8  1811.2  2648.3  6112.5 

42 

44 

8294.4   770.1  2573.9  6010.1 

8303.1  1812.4  2650.9  6116  1 

44 

46 

8296.7   777.3  2576.4  6013.6 

8305.4  1813.6  2053.5  6119  7 

46 

48 

8299.0   778.5  2578.9  6017.1 

8367.7  1814.9  2650.1  6123.2 

48 

50 

8301.3   779.7  2581.5  0020.6 

8309.9  1816.1  2058.7  61-J6.8 

50 

52 

8303.6   780.9  2584.0  0024.1 

8372.2  1817.3  2061.3  0130.4 

52 

54 

8305.9   782.1  2580.0  00-27.0 

8374.5  1818.5  2663.9  6133.9 

54 

5(5 

8308.2   7H3.3  2589.1  0031.1 

8376.8  1819.7  2000.0  6137.5 

56 

58 

S:J10.5   784.5  2"91.7  6034.6 

837!).!  1820.9  200!).  2  6141.1 

58 

GO 

S3  12.  8   785.7  2594  2  003S  •> 

H3S1  3  1S20.2  2J57I.8  OH  4.  7 

60 

292       IX.— FUNCTIONS  OF  A  ONE-DEGKEE  -CURVE. 


/ 

1)4° 

95° 

L.  C.    M.    E.    T. 

L.  C.   M.    E.     T. 

0 

8381.3  1822.2  2671.8  6144.7 

8449.2  1858.9  2751.5  6253.2 

0 

2 

8383.6  1823.4  2674.4  6148.3 

8451.5  1860.1  2754.2  6256.9 

2 

4 

8385.9  1824.6  2(577.0  6151.9 

8453.7  1861.3  2756.9  6260.5 

4 

6 

8388.1  1825.8  2679.7  6155.4 

8456.0  1862.6  2759.6  6264.2 

6 

8 

8390.4  1827.0  2(582.3  6159.0 

8458.2  1863.8  2762.3  6267.8 

8 

10 

8392.7  1828.3  2684.9  6162.6 

8460.4  1865.0  £765  0  6271.5 

10 

|2 

831)5.0  1829.5  2687.6  6166.2 

8462.7  1866.3  2767.7  6275.2 

12 

14 

8397.2  1830.7  2690.2  6169.8 

8464.9  1867.5  2770.4  6278.  S 

14 

16 

8399.5  1831.3  2692.8  (5173.4 

8467.2  1868.7  2773.1  6282.5 

16 

18 

8401.7  1833.1  2695.6  6177.0 

8469.4  1869.9  2775.8  6286.2 

18 

20 

8404.  C  1834.4  2698.1  6180.6 

8471.7  1871.2  2778.5  6289.8 

20 

2-2 

8406.3  1835.6  2700.8  6184.2 

8473.9  1872.4  2781.2  6293.5 

22 

24 

840S.5  1836.8  2703.4  6187.8 

8476.2  1873.6  2784.0  6297.2 

24 

26 

8410.8  1838.0  2706.1  6191.5 

8478.4  1874.9  2786.7  6300.9 

26 

28 

8413.1  1839.3  2708.7  6195.1 

8480.7  1876.1  2789.4  6304.6 

28 

30 

8415.3  1840.5  2711.4  6198.7 

8482.9  1877.3  2792.1  6308.2 

30 

32 

8417.6  1841.7  2714.0  0203.  8 

8485.1  1878.6  2794.9  6311.9 

32 

34 

8419.9  1842.9  2716.7  6205.9 

8487.4  1879.8  2797.6  6315.6 

34 

36 

8422.1  1844.2  2719.3  6209.5 

8489.6  1881.0  2S00.3  6319.3 

36 

38 

8424.4  1845.4  2722.0  6213.2 

8491.9  1882.3  2803.1  6323.0 

38 

40 

8426.6  1846.6  2724.7  6216.8 

8494.1  1883.5  2805.8  6326.7 

40 

42 

8428.9  1847.8  2727.3  6220.4 

8496.3  1884.8  2808.6  6330.4 

42 

44 

8431.2  1849.1  2730.0  6224.1 

8498.6  1886.0  2811.3  6334.1 

44 

46 

8433.4  1850.3  2732.7  6227.7 

8500.8  1887.2  2814.1  6337.8 

46 

48 

8435.7  1851.5  2735.4  6231.3 

8503.0  1888.5  2816.8  6341.5 

48 

50 

8437.9  1852.7  2738.0  6235.0 

8505.3  1889.7  2819.6  6345.2 

50 

52 

8440.2  1854.0  2740.7  6238.6 

8507.5  1890.9  2822.3  6349.0 

52 

54 

8442.4  1855.3  2743.4  6242.3 

8509.8  1892.2  2825.1  6352.7 

54 

56 

8444.7  1856.4  2746.1  6245.0 

8512  0  1893.4  2827.8  6356.4 

56 

58 

8447.0  1857.6  2748.8  6249.6 

8514.2  1894.6  2830.6  6360.1 

58 

60 

8449.2  1858.9  2751.5  6253.2 

8516.4  1895.9  2833.4  6363.8 

60 

96° 

97" 

/ 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     T. 

/ 

0 

8516.4  1895.  S  2833.4  6363.8 

8583.0  1933.2  2917.5  6476.6 

0 

2 

85187  1897.1  2836.1  6367.5 

8585.2  1934.4  2920.3  6480.4 

2 

4 

8520.9  1898.4  2838.9  6371.3 

8587.5  1935.7  2923.2  6484.2 

4 

6 

8523.1  1899.6  2841.7  6375.0 

8589  7  1936.9  2926.0  6488.0 

6 

8 

8525.4  1900.8  2844.5  6378.7 

8591.9  1938.2  2928.9  6491.8 

8 

10 

8527.6  1902.1  2847.2  6382.5 

8594  1  1939.4  2931.7  6495.6 

10 

12 

8529  8  1903.3  2850.0  6386.2 

8596.3  1940.7  2934.6  6499.4 

12 

14 

8532.0  1904.6  28528  6389.9 

859S.5  1941.9  2937.5  6503.2 

14 

16 

85343  1905.8  28556  6393.7 

8(500.7  1943.2  2940.3  6507.1 

16 

18 

8536.5  1907.0  2858.4  6397.4 

8602.9  1944.4  2943.2  6510.9 

18 

20 

8538.7  1908.3  2861.2  6401.2 

8605  1  1945.7  2946.1  6514.7 

20 

22 

8540  9  1909.5  2864.0  6404.9 

8607.3  1946.9  2948.9  6518.5 

22 

24 

8543.2  1910.8  2866.7  6408.7 

8609.5  1948.2  2951.8  6522.3 

24 

26 

8545.4  1912.0  2869.5  6412.4 

8611.7  1949.4  2954.7  6526.2 

26 

28 

8547.6  1913  3  2S72.3  6416.2 

8613.9  1950.7  2957.6  6530.0 

28 

30 

8549.8  1914.5  2875.1  6419.9 

86161  1952.0  2900.4  6533.8 

30 

32 

8552.0  1915.7  2877.9  6423.7 

8618.3  1953.2  2963.3  6537.7 

32 

34 

8554.3  1917.0  2880  8  6427.5 

8620.5  1954.5  2966.2  6541.5 

34 

36 

8556.5  1918.2  2883.6  6431.2 

8622.7  1955.7  2969.1  6545.3 

36 

38 

8558.7  1919.5  2886.4  6435.0 

8624.9  1957.0  2972.0  6549.2 

38 

40 

8560.9  1920.7  2889.2  6438.8 

8627.1  1958.2  2974.9  6553.0 

40 

42 

8563.1  1922.0  2892.0  64425 

8629.3  1959  5  2977.8  6556  9 

42 

44 

8565  3  1923.2  2894  8  6446.3 

8631.5  1960.7  2980.7  6500.7 

44 

46 

8567  6  1924  5  2897.7  6450.1 

8633.7  1962.0  2983.6  6564.6 

46 

48 

8569  8  1925.7  2900.5  6453.9 

8035.8  1963.2  2986.5  6568  4 

48 

50 

8572.0  1927.0  2903.3  6457.6 

8638.0  1964.5  2989.4  6572.3 

50 

52 

8574.2  1928.2  2906.!  (5461.4 

8640.2  1965.8  2992.3  6576.2 

52 

54 

8576.4  19-,'9  4  2!)09.0  6465.2 

8642.4  1967.0  2995  2  6580.0 

54 

56 

8578.6  1930.7  2!)11.8  15169.0 

HC.M.6  190S.3  299S.1  6583.9 

56 

58 

8580.8  1931.9  2914.7  6472.8 

8(546  -•>  !'.)()!).  5  3001.1  0587.7 

58 

60 

8583.0  1933.2  2917.5  6476.6 

80490  1970.8  30040  0591.6 

60 

IX.— FUNCTIONS  OF    A.  ONE-DEGREE  CURVE.      293 


98° 

99° 

/ 

I,.  C.    M.     E.    T. 

L.  C.    M.     E.    T. 

0 

8649.0  1970.8  3001.  0  6591.6 

8714.3  2008.7  3092.9  6709.0 

0 

2 

8051.2  1972.0  3006.9  6595.5 

8716.4  2009.9  S095.9  6712.9 

2 

4 

8053.3  1973.3  3009.8  6599.4 

8718.6  2011.2  3098.9  6716.9 

4 

6 

8655.5  1974.6  3012.8  6603.2 

8720.7  2012.5  3101.9  0720.8 

6 

8 

8657.7  1975.8  3015.7  6607.1 

1?722.9  2013.7  3104.9  6724.8 

8 

10 

8659.9  1977.1  3018.6  6611.0 

8725.1  2015.0  3107.9  6728.8 

10 

12 

8602.1  1978.3  3021.6  6614.9 

8727.2  2016.3  3111.0  6732.7 

12 

14 

8064.3  1979.6  3024.5  6618.8 

8729.4  2017.5  3114.0  6736.7 

14 

16 

8000.4  1980.9  3027.5  6022.7 

8731.5  2018.8  3117.0  6710.7 

16 

18 

8668.6  1982.1  3030.4  6026.  6 

8733.7  2020.1  3120.0  6744.6 

18 

'20 

8670.8  1983.4  3033.3  6030.5 

8735.9  2021.4  3123.1  6748.6 

20 

22 

8673  0  1984.6  3036.3  6634.4 

8738.0  2022.6  3126  1  6752.6 

22 

24 

8675.2  1985.9  3039.3  6638.3 

8710.2  2023.9  3129.1  6756.6 

24 

26 

8677.3  1987.2  3042.2  6612.2 

8712.3  2025.2  3132.2  6700.6 

26 

28 

8679  5  19P8.4  3045.2  00-10.1 

8744  5  2026.4  3135.2  6764.6 

28 

30 

8681.7  19S9.7  3048.1  6650.0 

8740.6  2027.7  3138.3  6708.6 

30 

32 

8683.9  1991.0  3051  1  0653.9 

8748.8  2029.0  3141.3  6772.6 

32 

34 

8080.0  1W2.2  3054.1  6657.8 

8750.9  2030.3  3144.4  6776.6 

34 

36 

8688  2  199H.5  3057.0  0601.7 

8753.1  2031.5  3147.4  67")0.6 

36 

38 

8690.4  1994.7  3060.0  6665.7 

8755.3  2032.8  3150.5  6781.6 

38 

40 

8692.6  1996.0  3063.0  6669.6 

8757.4  2034.1  3153.5  6788.6 

40 

42 

8094.7  1997.3  3006.0  6673.5 

8759.5  2035.4  3156.6  6792.6 

42 

44 

8696.9  1998.5  3068.7  6677.4 

8761.7  2036.6  3159.7  6790.6 

44 

46 

8699.1  199;).  8  3071.9  66S1.4 

8763.8  2037.9  3162.7  6800.6 

46 

48 

8701.2  2001.1  3074.9  6685.3 

8760.0  2039.2  3165.8  0804.6 

48 

50 

8703.4  2002.3  3077.9  6689.2 

8768.1  2040.5  3168.9  6808.6 

50 

52 

8705.6  2003.6  3080.9  0693.2 

8770.3  2041.7  3172.0  6812.6 

52 

54 

8707.8  2004.9  3083.9  6697.1 

8772.4  2043.0  HIT5.1  OS10.7 

54 

56 

8709.9  2006.1  3086.9  0701.1 

8774.6  2044.3  3178.1  6820.7 

56 

58 

8712  1  2007.4  3089.9  6705.2 

8776.7  2045.6  3181.2  6824.7 

58 

CO 

8714.3  2008.7  3092.9  67'09.0 

8778.9  2046.8  3184.3  6828.8 

60 

100° 

101° 

L  C.    M.     E.    T. 

L.  C.    M.     E.    T. 

0 

8778.9  2046.8  3184.3  6828.8 

8842.8  2085.3  3278.3  6951.0 

0 

8781.0  2048.1  3187.4  0832.8 

8S44.9  2086.6  3281.5  6955.2 

2 

4 

8783.1  2049.4  3190.5  C830.8 

8847.0  2087.8  3284.7  6959  3 

4 

6 

8785.3  2050.7  3193.6  0840.  9 

8849.2  2089.1  3287.9  6963.4 

6 

8 

8787  4  2051.9  3196.7  6844.9 

8851.3  2090.4  3291.1  6967.6 

8 

10 

8789.6  2053.2  3199.8  6849.0 

8853.4  2091.7  3294.3  6971.7 

10 

13 

8791.7  2054.5  3202.9  6853.0 

8855.5  2093.0  3297.5  6975.8 

12 

14 

8793.9  2055.8  3206.0  6857.1 

8857.6  2094.3  3300.7  6980.0 

14 

16 

8796.0  2057.1  3209.1  6861.1 

8859.8  2095.6  3303.9  0984.1 

16 

18 

8798.9  2058.3  3212.2  6865.2 

8861.9  2096.9  3307.1  6988.2 

18 

20 

8800  3  2059.6  3215.4  6869.2 

8864.0  2098.2  3310.3  6992  4 

20 

22 

8802.4  2060.9  3218.5  6873.3 

8866.1  2099.4  3313.5  6996.6 

22 

24 

8804.5  2062.2  3221.6  6877.4 

8868.2  2100.7  3316.7  7000  7 

24 

26 

8S06.7  2063.5  3224.7  6881.4 

8870.3  2102.C  3319.9  7004.9 

26 

28 

8808.8  2064.7  3227.9  0885.5 

8872.4  2103.3  3323.1  70090 

28 

30 

8810  9  2066.0  3231  0  6889.6 

8874.5  2104.6  3326  4  7013.2 

30 

32 

8813.1  2067.3  3234.1  6893.7 

8876.7  2105.9  3329.6  7017.3 

32 

34 

8815.2  2068.6  3237  3  6897.8 

8878.8  2107.2  3332.8  7021  5 

34 

36 

8817.3  2069  9  3240.4  6901.8 

8880.9  2108.5  3336  0  7025.7 

36 

38 

8819.5  2071.1  3243.5  6905.9 

8883.0  2109.8  3339.3  7029.9 

38 

40 

8821.6  2072.4  3246.7  6910.0 

8885.1  2111.1  3342.5  7034.0 

40 

42 

HS23.7  2073.7  3249.8  6914.1 

8887.2  2112.4  3345.8  7038.2 

42 

44 

8825.8  2075.0  3253.  C  6918.2 

8889.3  2113.6  3349.0  7042.4 

44 

46 

8828.0  2076.3  3250.2  6922.3 

8891  4  21149  3352.3  7046.6 

46 

48 

8830.1  2077.6  3259.3  6926.4 

8893.5  2116.2  3355.5  7050.8 

48 

50 

8832.2  2078.9  3262.5  6930.5 

8895.  6  2117.5  3358  8  7055  0 

50 

52 

8834.3  2080.1  3265.7  6934.6 

8897.7  2118.8  3362.0  7059.2 

52 

54 

8S36.4  2081.4  3268.8  6938.7 

8899.8  2120.1  3365  5  7063  4 

54 

56 

8838  6  2082.7  3272  0  6942.8 

8901.9  2121.4  3368.7  70076 

56 

58 

S3  10  7  20H4.0  3275.2  6940.9 

8904.0  2122.7  3372.0  7071.8 

58 

60 

88428  2085.3  3278.3  6951.0 

8906.1  2124.0  3375.1  7076.0 

60 

294    IX.— FUNCTIONS  OF  A   ONE-DEGREE  CURVE. 


1O2° 

103° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.     T. 

0 

890(5.1  21240  3375.1  70760 

8968.7  2163.0  3474.6  7203.6 

0 

2 

8908.2  2125.3  3378.3  7080.2 

8970.8  2164.3  3478.0  7207.9 

2 

4 

8910.3  2126.6  33S1.6  7084.4 

8972.9  2165.6  3481.4  1212.3 

4 

6 

8912  4  2127.9  3384.9  7088.6 

8974.9  2166.9  3484.7  7216  5 

6 

8 

8914.5  2129.2  3388.2  7092.8 

8977.0  2168.2  3488.1  7220.8 

8 

10 

8916.6  2130.5  3391.5  7097.1 

8979.1  2169.5  3491.5  7225.1 

10 

18 

8918.7  2131.8  3394  7  7101.3 

8981.1  2170.8  3494.9  7229.5 

12 

14 

8920.8  2133.1  3398.0  7105.5 

8983.2  2172.1  3498.3  7233.8 

14 

16 

8922.9  2134.4  3401.3  7109.7 

8985.3  2173.4  3501.6  7238.1 

16 

18 

8925.0  2135.7  3104.6  7114.0 

8987.3  2174.7  3505.3  7242.4 

18 

20 

8927.0  2137.0  3407.9  7118.2 

8989.4  2176.1  3508.4  7246.8 

20 

22 

8929.1  2138.3  3411.2  7122.4 

8991.5  2177.4  3511.8  7251.1 

22 

24 

8931.2  2139.6  3414.5  7126.7 

8993.5  2178.7  3515.2  7255  4 

24 

26 

8933.3  2140.9  3417.9  7130.9 

8995.6  2180.0  3518.7  7259.  s 

26 

28 

89354  2142.2  3421.2  71352 

8997.7  2181.3  3522.1  7264.1 

28 

30 

8937.5  2143.5  3424.5  7139.4 

8999.7  2182.6  3525.5  7268.5 

30 

32 

8939.6  2144  8  3427.8  7143.7 

9001.8  2183.9  3528.9  72728 

32 

34 

8941.6  2146.1  3431.1  7148.0 

9003.9  2185.2  3532.3  7277.2 

34 

36 

8943.7  2147.4  34:34.5  7152.2 

9005.9  2186.5  3535.7  728  1.5 

36 

38 

8945.8  2148.7  3437.8  7156.5 

9008.0  2187.8  3539.2  7285.9 

38 

40 

8947.9  2150.0  3441.1  7160.7 

9010.0  2189.1  3542.6  7290.3 

40 

42 

8950.0  2151.3  3444.4  7165.0 

9012.1  2190.5  3546.0  7294.6 

42 

44 

8952.1  2152.6  3447.8  7169.3 

9014.2  2191.8  3549.5  7299.0 

44 

46 

8954.1  2153.9  3451.1  7173.6 

9016.2  2193.1  3552.9  7303.4 

46 

48 

8956.2  2155.2  3454.5  7177.9 

9018.3  2194.4  3556.3  7307.7 

48 

50 

8958.3  2156.5  3457.8  7182.1 

9020.3  21957  3559.8  7312.1 

50 

52 

8960.4  2157.8  3461.2  7186.4 

9022.4  2197.0  356:5.2  7316.5 

52 

54 

8962.5  2159.1  3464.5  7190.7 

9024.5  2198.3  3566.7  7320.9 

54 

56 

8964.5  2160.4  3467.9  7195.0 

9026  5  2199.6  3570.2  7325.3 

56 

58 

K966.6  2161.7  3471.2  7199.3 

9028.6  2200.9  3573.6  7329.7 

58 

60 

8968.7  2163.0  3474.6  7203.6 

9030.6  2202.3  3577.1  7334.1 

60 

104° 

105° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

9030.6  2202.3  3577.1  7334.1 

9091.8  2241.8  3682.6  7467.5 

0 

2 

9032.7  2203.6  3580.5  7338.5 

9093.9  2243.1  3686.1  7472.0 

2 

4 

9034.7  2204.9  3584.0  7342.9 

9095.9  2244.4  3689.7  7476.5 

4 

6 

9036.8  2206.2  3587.5  7347.3 

9097.9  2245.8  3693.3  7481.0 

6 

8 

'J038.8  2207.5  3591.0  7351.7 

9099.9  2247.1  3696.9  7485.5 

8 

10 

9040.9  22088  3594.4  7356.1 

9102.0  2248.4  3700.4  7490.0 

10 

12 

9042.9  22102  3597.9  7360.5 

9104.0  2249.7  3704.0  7494.5 

12 

14 

9045.0  2211.5  3601.4  7364.9 

9106.0  2251.1  3707.6  7499.1 

14 

16 

9047.0  2212.8  3604.9  7369.4 

9108.0  2252.4  3711.2  7503.6 

16 

18 

9049.1  2214.1  3608.4  7373.8 

9110.1  2253.7  3714.8  7508.1 

18 

20 

9051.1  2215.4  3611.9  7378.2 

9112.1  2255.0  3718.4  7512.6 

20 

22 

9053.1  2216.7  3615.4  7382.6 

9114.1  2256.4  3722.0  7517.2 

22 

24 

9055.2  2218.0  3618.9  7387.1 

9116.1  2257.7  3725.6  7521.7 

24 

26 

9057.2  2219.4  3622.4  7391.5 

9118.1  22.19.0  3729.3  7526.3 

26 

28 

9059.3  2220.7  3625.9  7396.0 

9120.2  2260.3  3732.9  7530.8 

28 

30 

9061.3  2222.0  3629.4  7400.4 

9122.2  2261.7  3736.5  7535.3 

30 

32 

9063.3  2223.3  3633.0  7404.8 

9124.2  2263.0  3740.1  7539  9 

32 

34 

9065.4  2224.6  3636.5  7409.3 

912B.3  2264.3  3743.7  7544.4 

34 

36 

9007.4  2226  0  3640.0  7413  8 

91282  2205.7  3747.4  7519.0 

36 

38 

9069.5  2227.3  3643.5  7418.2 

91302  2267.0  3751.0  75536 

38 

40 

9071.5  2228.6  3647.1  7422.7 

9132.3  2268.3  3754.6  7558.1 

40 

42 

90735  2229.9  3650.6  7427.1 

9134.3  2269.6  3I5S.3  7562.7 

42 

44 

90756  2231.2  3654.1  7431.6 

9136.3  2271.0  3761.9  7567.3 

44 

46 

9077  6  2232.6  3657.7  7436.1 

9138.3  2272.3  3765.6  7571.8 

46 

48 

9079.6  2233.9  3661.2  7440.6 

9140.3  2273  6  3769.2  7576.4 

4S 

50 

9081.7  2235.2  3664.8  7445.0 

9142.3  2275.0  3772.9  7581.0 

50 

52 

9083.7  2236.5  3668.3  7449.5 

9144.3  2276.3  3776.5  7585.6 

52 

54 

9085.7  2237.8  3671.9  7454.0 

9146.3  2277.6  3780.2  7590.2 

54 

56 

90S7.8  2239.2  3675.4  7458  5 

9148.3  2278.9  3783.9  7594.8 

56 

66 

0089.8  2240.5  3679.0  7463.0 

9150.4  2280.3  3787.5  7599.4 

58 

60  190918  2241.  K  :',<>S2  fi  741575 

9152  4  22S1.fi  3701.2  7W4  0 

60 

IX.— FUNCTIONS   OP   A   ONE  DEGREE   CURVE.     295 


106° 

107° 

L  C.   M.    E.     T. 

L.  C.    M.    E.     T. 

0 

9152.4  2-281.6  3791.2  7604.0 

9212.2  2321.7  3903.1  7743.7 

0 

2 

9151.4  2282.9  3794.9  7608.6 

9214.2  2323.0  3906.9  7748.4 

2 

4 

9156.4  2284.3  3798.6  7613.2 

9216.2  2324.4  3910.7  7753.1 

4 

6 

9158.4  2285.6  3802.3  7617.8 

9218.1  2325.7  3914.5  7757.8 

6 

8 

9160.4  22S6.9  3805.9  7622.4 

9220.1  2327.0  3918.3  7762.5 

8 

10 

9162.4  2288.3  3809.6  7627.0 

9222.1  2328.4  3922.1  7767.3 

10 

12 

9184.4  2289.6  38133  7631.7 

9224.1  2329.7  3925.9  7772.0 

12 

14 

9166.4  2290  9  3817.0  7630.3 

9226.1  2331.1  3929.7  7776.7 

14 

16 

9168.4  2292.3  3820.7  7610.9 

9228.1  2332.4  3933.6  7781.5 

16 

18 

9170.4  2293.6  3824.4  7645.5 

9230.0  2333.7  3937.4  7786.2 

18 

20 

9172  4  2294.9  3828.1  7650.2 

9232.0  2335.1  3941.2  7791.0 

20 

22 

9174.4  2296.3  3831.9  7654.8 

9234.0  2336.4  3945.0  7795.7 

22 

24 

9176.4  2297.6  3835.6  7659.5 

9235.9  2337.8  3948.9  7800.5 

24 

26 

9178.4  2298.9  3839.3  7664.1 

9237.9  2339.1  3952.7  7805.2 

26 

28 

9180  4  2300.3  3843.0  7668.8 

9239.9  2340.5  3956.5  7810.0 

28 

30 

9182.4  2301.6  3846.7  7673.4 

9241.9  2341.8  3960.4  7814.7 

30 

32 

9184.4  2302.9  3850.5  7678.1 

9243.8  2343.1  3964.2  7819.5 

32 

34 

9186.4  2304.3  3854.2  7682.7 

9245.8  2344.5  3968.1  7824.3 

34 

36 

9188.4  2305.6  3858.0  7687.4 

9247.8  2345.8  3971.9  7829.1 

36 

38 

9190.4  2306.9  3861.7  7G92.1 

9249.7  2347.2  3975.8  7833.8 

38 

40 

9192.4  2308.3  3865.4  7696.7 

9251.7  2348.5  3979.6  7838.6 

40 

42 

9194.4  2309.6  3869.2  7701.4 

9253.7  2349.9  3983.5  7843.4 

42 

44 

9196.3  2311.0  3873.0  7706.1 

9255.6  2351.2  3987.4  7848.2 

44 

46 

9198.3  2312.3  3876.7  7710.8 

9257.6  2352.6  3991.3  7853.0 

46 

48 

9200.3  2313.6  3880.5  7715  5 

9259.6  2353.9  3995.1  7857.8 

48 

50 

9202.3  2315.0  3884.2  7720.1 

9261.5  2355.3  3999.0  7862.6 

50 

52 

9204  3  2316.3  3888.0  7724.8 

9203.5  2356.6  4002.9  7867.4 

52 

54 

92063  2317.7  3891.8  7729.5 

9265.4  2358.0  4006.8  7872.2 

54 

56 

9208.2  2319.0  3895.6  7734.2 

9207.4  2359.3  4010.7  7877.0 

56 

58 

9210.2  2320.3  3899.3  7739.0 

9269.4  2300.7  4014.6  7881.9 

58 

60 

9212.2  2321.7  3903.1  7743.7 

9271.3  2362.0  4018.5  7886.7 

60 

108° 

109° 

/ 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

/ 

0 

9271.3  2362.0  4018.5  7886.7 

9329.8  2402.6  4137.4  8033.2 

0 

2 

9273.3  2363.3  4022.4  7891.5 

9331.7  2403.9  4141.4  8038.1 

2 

4 

9275.3  2364.7  4026.3  7896.3 

933.-].  6  2405.3  4145.4  8043.1 

4 

6 

9277.2  2366.0  4030.2  7901.2 

9335.6  2406.6  4149.5  8048.0 

6 

8 

9279.2  2367.4  4034.1  7906.0 

9337.5  2408.0  4153.5  8053.0 

8 

10 

9281.1  2368.7  4038.0  7910.8 

9339.4  2409.4  4157.5  8057.9 

10 

12 

9283.1  23701  4042.0  7915.7 

9341.4  2410.7  4161.6  8062.9   12 

14 

9285.0  2371.4  4045.9  7920.5 

9343.3  2412.1  4165.6  8067.9 

14 

16 

9287.0  2372.8  4049.8  7925.4 

9345.2  2413.4  4169.7  8072.8 

16 

18 

9288.9  2374.1  4053.8  7930.3 

9347.2  2414.8  4173.8  8077.8 

18 

20 

9290.9  2375.5  4057.7  7935.1 

9349.1  2416.2  4177.8  8082.8 

20 

22 

9292.8  2376.8  4061.6  7940.0 

9351.0  2417.5  4181.9  8087.8 

22 

24 

9294.8  2378.2  4065.6  7944.8 

9352.9  2418.9  4186.0  8092.8 

24 

26 

9296.7  2379.5  4069.5  7949.7 

9354.9  2420.2  4190.0  8097.8 

26 

28 

9298.7  2380.9  4073.5  7954.6 

9356.8  2421.6  4193.1  8102.8 

28 

30 

9300.6  2382.3  4077.5  7959  5 

9358.7  2423.0  4198.2  8107.8 

30 

32 

9302.6  2383.6  4081.4  7964.4 

9360.6  2424.3  4202.3  8112.8 

32 

34 

9304.5  2385.0  4085.4  7969.3 

9362.6  2425.7  4206.4  8117.8 

34 

36 

9306.5  2386.3  4089.4  7G74.1 

9364.5  2427.0  4210.5  8122.8 

36 

38 

9308.4  2387.7  4093.4  7979.0 

9366.4  2428.4  4214.6  8127.8 

38 

40 

9310.4  2389.0  4097.3  7983.9 

9368.3  2430.0  4218.7  8132.8 

40 

42 

9312.3  2390.4  4101.3  7988.8 

9370.2  2431.1  4222.8  8137.9 

42 

44 

9314.2  2391.7  4105.3  7993.8 

9372.2  2432.5  4226.9  8142.9 

44 

46 

9316.2  2393.1  4109.3  7998.7 

9374.1  2433.9  4231.0  8147.9 

46 

48 

9318.1  2394.4  4113.3  8003.6 

9376.0  2435.2  4235.1  8153.0 

48 

50 

9320.1  231)5.8  4117.3  8008.5 

9377.9  2436.6  42393  8158.0 

50 

52 

9322.0  2397.2  4121.3  8013.4 

9379.8  2438:0  4243.4  8163.1 

52 

54 

9323  9  2398.5  4125.3  8018.4 

9381.7  2439.3  4247.5  8168.1 

54 

56 

9325.9  2399.9  4129.3  8u23  3 

9383.7  2440.7  4251.7  8173.2 

56 

58 

9327.8  2401.2  4133.4  8028.2 

93S5.6  2442.1  4255.8  8178.2 

58 

60 

932!l.8  2402.6  4137.4  8033.2 

93S7.5  2443  4  4260.0  8183.3   60 

296    IX.— FUNCTIONS  OF  A   ONE-DEGREE  CURVE. 


110° 

111° 

1 

L.  C.    M.     E.    T. 

L.  C.    M.    E.     T. 

'   1 

0 

9387.5  2143.  4  42(50.0  8!  S3.  3 

9444.5  2484.5  4386.4  8337.2 

0 

2 

9389.4  2444.8  4-J64.1  8188.4 

9446.4  2485.9  4390.7  8342  4 

2 

4 

9391.3  2446.1  4208.  3  8193.4 

9448.3  2487.2  4395  0  83  47.  (5 

4 

6 

9393  2  2447.5  4272.4  8198.5 

9450.1  2488.6  4399.3  8352.8 

6 

8 

9395.1  2448.9  4276.6  8203.6 

9452.0  2490.0  4403.6  8358  0 

8 

10 

9397.0  2450.2  4280.8  8208  7 

9453.9  2491.4  4407.9  8:503.2 

10 

12 

9398.9  2451.6  4284.9  8213.8 

9455.8  2492.7  4412.2  8368.5 

12 

14 

9400.8  2453.0  4289.1  8218.9 

9457.7  2494.1  4416.5  8373.7 

14 

16 

9402.7  2454.3  42<>3.3  8224.0 

9459.6  2495.5  4420.8  8378  9 

16 

18 

9404.7  2455.7  4297.5  8229.1 

9461.4  2496.9  4425.1  8384  1 

18 

20 

9406.6  2457.1  4301.7  8234.2 

9463  3  2498.2  4429.5  8389.4 

20 

22 

9408.5  2458.4  4305.9  8239.3 

9165.2  2499.6  4433.8  8394  6 

22 

24 

9410.4  2459  8  4310.1  8244.4 

9467.1  2504.0  4438.1  83S9.9 

24 

26 

9412.3  2461.2  4314.3  8249.5 

9469.0  2502.4  4442.5  8405.1 

26 

28 

9414.2  2462.6  4318.5  8254.6 

9470.8  2503.8  4446.8  8410.4 

28 

30 

9416.1  24(53.9  4322.7  8259  8 

9472.7  2505.1  4451.2  8415.6 

30 

32 

9418.0  2465.3  432(5.9  8264.9 

9474.6  2506.5  4455.5  8420.9 

32 

34 

9419.9  2466.7  4331.1  8270.0 

9476.5  2507.9  4459.9  8426.2 

34 

36 

9421.8  2468.0  4335.4  8275.2 

9478.3  2509.3  4464.2  8431.4 

36 

38 

9423.7  2469.4  4339.6  8280.3 

9480.2  2510.6  4468.6  8436.7 

38 

40 

9425.6  2470.8  4343.8  8285.5 

9482.1  2512.0  4473.0  8442.0 

40 

42 

9427.5  2472.1  4348.1  8290.6 

9484.0  2513.4  4477.3  8447.3 

42 

44 

9429.3  2473.5  4352.3  8295.8 

!»4S5  8  2514.8  4481.7  8452.6 

44 

46 

9431.2  2474.9  4356.6  8300.9 

9487.7  2516.2  4486.1  8457.9 

46 

48 

9433.1  2476.3  4360.8  8306.1 

9489.6  2517.5  4490.5  8463.2 

48 

50 

9435.0  2477.6  4365.1  8311.3 

9491.4  2518.9  4494.9  8468.5 

50 

52 

9436.9  2479.0  4369.3  8316.5 

9493.3  2520.3  4499.3  8473.8 

52 

54 

9438.8  2480.4  4373.6  8321.6 

9495.2  2521.7  4503.7  8479.1 

54 

56 

9440.7  2481.7  4377.9  8326.8 

9497.0  2523.1  4508.1  8484.4 

56 

58 

9442.6  2483.1  43822  8332.0 

9498.9  2524.5  4512.5  8489.7 

58 

60 

9444.5  2484  5  4380.4  8337.2 

9500.8  2525.8  4516.9  8495.1 

60 

112° 

113° 

L.  C.   M.    E.     T. 

L.  C.   M.     E.     T. 

0 

9500.8  2525.8  4516.9  8495.1 

9556.3  2567.4  4651.6  8657.1 

0 

2 

9502.6  2527.2  4521.4  8500.4 

9558.2  2568.8  4656.2  8602.6 

2 

4 

9504.5  2528.6  4525.8  8505.8 

9560.0  2570.2  4660.8  8668.0 

4 

6 

9506.4  2530.0  4530.2  8511.1 

9561.8  2571.6  4665.4  8673.5 

6 

8 

9508.2  2531.4  4534.6  8516.4 

9563.7  2573.0  4669.9  8679.0 

8 

10 

9510.1  2532.7  4539.1  8521.8 

9565.5  2574.4  4674.5  8684.5 

10 

12 

9511.9  2534.1  4543.5  8527.1 

9567.4  2575.8  4679.1  8690.0 

12 

14 

9513.8  2535.5  4548.0  8532.5 

9569.2  2577.1  4683.7  8695.5 

14 

16 

9515.7  .2536.9  4552.4  8537.9 

9571.0  2578.5  4(588.3  8701.0 

16 

18 

9517.5  2538.3  4556.9  8543.2 

9572.9  2579.9  4692.9  8706.5 

18 

20 

9519.4  2539.7  4561.3  8548.6 

9574.7  2581.3  4697.5  8712.0 

20 

22 

9521.2  2541.0  4565.8  8554.0 

9576.5  2582.7  4702.1  8717.6 

22 

24 

9523.1  2542.4  4570.3  8559  4 

9578.4  2584.1  4706.8  8723.1 

24 

26 

9524  9  2543.8  4574.8  8564.8 

95802  2585.5  4711.4  8728.6 

26 

28 

9526.8  2545.2  4579.3  8570  2 

9582.0  2586.9  4716.0  8734.2 

28 

30 

9528.6  2546.6  4583.7  8575.6 

9583  8  2588.3  4720.6  8739.7 

30 

3S 

95305  2548.0  4588.2  8581.0 

9585  7  2589.7  4725.3  8745.3 

32 

¥4 

9:.32.3  2549.4  4592.7  8586.4 

9587.5  2591.1  4729.9  8750.8 

34 

81 

9534.2  2550.7  4597.2  S591.8 

9589.3  2592.5  4734.6  8756.4 

36 

3S 

95360  2552.1  4C01.7  8597.2 

951)1.1  2593.9  4739.2  8761.9 

38 

40 

9537.9  2553.5  4606  2  8602.6 

9593.0  2595.3  4743.9  8:157.5 

40 

42 

9539.7  2554.9  4610  8  8608.0 

9594.8  2596.7  4748.5  8773.1 

42 

44 

9541.6  2556.3  4615.3  (-6I3.5 

!>5'.)6.6  2598.1  4753.2  8778.6 

44 

46 

9543.4  2557.7  4619.8  8018.9 

9598.4  25994  4757.9  87S1.2 

46 

48 

9545.3  2559.1  4624.3  8624.3 

9600.3  2600.8  4762.6  8789.8 

48 

50 

9547.1  2560.5  4628.9  8629.8 

9602.1  2602.2  4767.2  8795.4 

50 

52 

9S.49.0  2561.8  4633.4  8(535.2 

9603.9  2603.6  4771.9  8801.0 

52 

54 

9550.8  2563  2  4638.0  8640.7 

9605.7  2605  0  4776.6  8806.6 

54 

56 

9552.6  2564.6  4642.5  8646.2 

9607.5  2606.4  4781.3  8812.2 

56 

58 

{'554.5  2566.0  4647  1  8651.6 

9609.4  2607.8  4786.0  8817.8 

58 

60 

95563  2567.4  4651.6  8657.1 

9611.2  2609.2  4790.7  8823.4 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     297 


114° 

115° 

' 

L.  C.    M.    E.    T. 

L.  C.    M.    E.    T. 

0 

9611  2  2609.2  4790.7  8823.4 

9665.3  2651.3  4934.4  8994.3 

0 

2 

9(513.0  2610.6  4795.5  8829.1 

9667  1  2652.7  4939  3  9000.1 

2 

4 

9614.8  2612.0  4800.2  8834.7 

9668.8  2654.1  4944.2  9005.9 

4 

6 

9616.6  2613.4  4804.9  8840.3 

9670.6  2655.5  4949.1  9011.6 

6 

8 

9618.4  2614.8  4809.6  8846.0 

9672.4  2656.9  4954.0  90  17.  4 

8 

10 

9620.2  2616,2  4814.4  HS51.6 

9674.2  2658.3  4958.9  9023.2 

10 

13 

9622.0  2617.6  4819.1  8857.2 

9676.0  2659.7  4963.8  9029.0 

12 

14 

9623  8  2619.0  4823.9  8862.9 

9G77.8  2(361.1  4968.7  9034.8 

14 

16 

%25.7  2620.4  4828.6  8868.5 

9679.6  266'2.5  4973.6  9040.7 

16 

18 

9627.5  2621  8  4833.4  8874.2 

9681.4  2663.9  4978.5  9046.5 

18 

20 

9629.3  2623.2  4838.1  8879.9 

9683.1  2665.4  4983.4  90523 

20 

22 

9631,1  2624.6  4842.9  8885.5 

9684.9  2666.8  4988.3  9058.1 

22 

24 

9632.9  2626.0  4847.7  8891.2 

9686.7  266S.2  4993.3  9064.0 

24 

20 

9634.7  2627.4  4852.4  8896.9 

9688.5  2669  6  4998.2  9069.  H 

2(5 

28 

9636.5  26288  4857.2  «)02.« 

9690.3  2671.0  5003.2  9075.7 

28 

30 

96:^8.3  2630.2  4862.0  8908.3 

9692.0  2672.4  5008.1  9081  5 

30 

32 

9640.1  2631.6  4866.8  8914.0 

9693.  S  x!673.8  5013.1  9087.4 

32 

34 

9641.9  2633.0  4871.6  89197 

9695.  (5  2675.2  5018.0  9093.2 

34 

36 

9643.7  2634.4  4876.4  8K25.4 

W97.4  2676.6  5023.0  90119.1 

36 

38 

9645.5  2635.8  4881.2  8931.1 

9699.1  2678.0  5028.0  91050 

38 

40 

9<i»7.3  2637.2  4885.0  8936.8 

9700.9  2679.5  5032.9  9110.8 

40 

42 

1KM9.1  2638.6  4890.9  8942.6 

9702.7  2680.9  5037.9  9116.7 

42 

44 

9650.9  2640.0  4895.7  8948.3 

9704  5  2682.3  5042.9  9122.6 

44 

46 

9652.7  2641.4  4900.5  8954  0 

9706.2  2683.7  5047.9  9128.5 

46 

4H 

9654.5  2642  9  4905.3  8959  8 

9708.0  2685.1  5052.9  9134  4 

4S 

50 

9656.3  2644  3  4910.2  8965.5 

9709.8  2686.5  5057.9  9140.3 

50 

9658.1  2645.7  4915.0  8971.3 

9711.0  2687.9  5062.9  9146.2 

52 

51 

!»(>.->!).  9  2647.1  4919  9  8977.0 

9713.3  2689  3  5067.9  9152  1 

54 

56 

9(561.7  2648.5  4924.7  8982.8 

9715.1  261)0.7  5072.9  9158.1 

56 

58 

9663.5  2649.9  4929.6  P988.5 

9716.9  2692.2  5078.0  9164.0 

58 

60 

9665.3  26513  4934.4  8994.3 

9718.6  2693.6  5083.0  9169.9 

60 

lift0 

117° 

L.  C.    M.    E.     T. 

L.  C.    M.    E.    T. 

0 

9718.6  2693.6  5083.0  9169.9 

9771.3  2736.1  5236.6  9350.5 

0 

9720.4  2695.0  50880  9175.9 

9773.0  2737.5  5241.8  9356.6 

2 

4 

9722.2  2696.4  5093.1  9181.8 

9774.7  2738.9  5247.0  9362.7 

4 

6 

9723.9  2697.8  5098.1  9188.8 

9776.5  2740.4  5252.2  9368.9 

6 

8 

9725.7  2699.2  5103  2  9193.7 

97782  2741.8  5257.4  9375.0 

8 

10 

9727.4  2700.6  5108.2  9199.7 

9779.9  2743.2  5262.6  9381.1 

10 

12 

9729.2  2702.1  5113.3  9205.6 

97.81.7  27'44.6  5267.9  9387.3 

12 

14 

9731.0  2703.5  5118.4  9211.6 

9183  4  2746.0  5273.1  9393.4 

14 

16 

9732.7  2704.9  5123.4  9217.6 

9785.2  2747.5  5278.4  9399.5 

16 

18 

9734.5  2706.3  5128.5  92-.'3.6 

9786.9  2748  9  5283.6  9405.7 

18 

20 

9736.3  2707.7  5133.6  9229  6 

9788.6  2750.3  5288.9  9411.9 

20 

22 

9738.0  2709.1  5138.7  9235  5 

9790.4  2751.7  5294.2  9418.0 

22 

24 

9739.8  2710.6  5143.8  9211.5 

9792.1  27532  5299.5  9424.2 

24 

26 

9741.5  2712.0  5148.9  9247  6 

9793.8  2754.6  5304.7  9430.4 

26 

28 

9743.3  2713.4  51540  9253.6 

9795  6  2756.0  5310.0  9436.6 

28 

30 

9745.0  2714  8  5159.1  9250.6 

9797.3  2757.4  5315.3  9442.8 

30 

3'2 

9746.8  2716.2  5164.2  9265.6 

9790.0  2758.9  5320.6  9449.0 

32 

34 

9748.5  2717.6  5169.4  9271.6 

9800.  7  2760.3  5325.9  9455.2 

34 

36 

9750.3  2-719.1  5174.5  9277.7 

9802.5  2761.7  5331.2  9461.4 

36 

38 

9752.0  2720.5  5179.7  9283.7 

9804.2  2763.1  5336.5  9467.6 

38 

40 

9753.8  2721.9  5184.8  9289.8 

9805.9  2764.6  5341.8  9473.8 

40 

42 

9755.6  2723.3  5190.0  9295.8 

9807.7  2766.0  5347.2  9480.0 

42 

44 

9757.3  2724.7  5195.1  9301.9 

9809.4  5767.4  5352.5  9-186.3 

44 

46 

9759.0  2726.2  5200.3  9307.9 

9811.1  2768.8  5357.9  9192.5 

46 

48 

9760.8  2727.6  5205.4  9314.0 

9812.8  2770.3  5363.2  9498.7 

48 

50 

9762.5  2729.0  5210.6  9320.1 

9814.5  2771.7  5368.5  9505.0 

50 

52 

9764.3  2730.4  5215.8  9326.1 

9816.3  2773.1  5373.9  9511.2 

52 

54 

9766.0  2731.8  5221.0  9332.2 

9818.0  2774.6  5379  3  9517.5 

54 

56 

9767.8  2733.3  5226.2  933S.3 

9819.7  2776.0  53S4.7  9523.8 

56 

58 

9769.5  2734.7  5231.4  9314.4 

9821.4  2777.4  5390.0  9530.0 

58 

60 

9771.3  2736.1  5236.6  9350.5 

9823.1  2778.8  5395.4  9536.3 

60 

298 


TAHLR  x.-- SINES  AND  COSINES. 


0° 

1° 

2° 

3°  ,  I!    4° 

Sine  (Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  |  Cosin 

0 

.00000 

One. 

.01745 

.99985 

.08490 

.99939 

.05234 

.09868 

.00976  .99756 

60 

1 

.1.00891  One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99801 

.07005  .99754 

59 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860 

.070J34 

.99752 

58 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858! 

.0706.3 

.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.03000 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.05379 

.99855 

.07121 

.99740 

55 

6 

.00175 

One. 

.01920 

.99982 

.03604 

.99933 

!  05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.99981 

.03093 

.99932 

.05437 

.99852 

.07179 

.99742 

5:3 

8 

.00238 

One. 

.01978 

.99980 

.03723 

.99931 

.05406 

.99851 

.07208 

.99740 

52 

9 

.00262 

One. 

.02007  .99980 

.03752 

.99930 

.05495 

.99849 

.07237 

!  99738 

51 

lO 

.00291 

One. 

.02036  .99979  .03781 

.99929 

.05524 

.99847 

.07206 

.99736 

50 

11 

.00320 

.99999 

.02065  .99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00319 

.99999 

.02094 

.99978 

.03839 

.99920 

.05582 

.9984-1 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.0:3808 

.!)!)925 

.05611 

.9984:2 

.07353 

!  99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05069 

.99831) 

.07411 

.99725  45 

16 

.  00465  !.  99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723  44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.057'27 

.99830 

.07469 

.997'21 

43 

18 

.00524 

.99999 

.02209 

.99974 

.04013 

.99919 

.05756 

,99884 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.9997'4 

.04042 

.99918 

.05785 

.99833 

.077)27 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972  .04129 

.99915 

.05873 

.99827 

.07614 

.99710 

38 

23 

.00609  .99998 

.02414 

.99971 

.04159 

.99913 

.05902 

.9982(5 

.(.7643 

.99708  37 

24 

.00698 

(jnqnij 

.02443 

.99970 

!.  04188 

.99912 

.05931 

.99824 

.07(572 

.99705  ,36 

25 

.00727 

!  99997 

.0247'2 

.99969  1.04217 

.99911 

.05960 

.99822 

.07701 

.99703!  35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701  34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819 

.07759 

.99699  33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817 

.07788 

.99090  32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815  .07817 

.99094  31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813  .07846 

.99092 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02076 

.99964 

.04420 

.99902  .06163 

.99810  i  .07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901  1  .06192 

.99808 

.07933 

.99085 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.079(52 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680  25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99897 

.00279 

.99803 

.08020 

.99678  24 

37 

.010761.99994 

.02821 

.99960 

.04565 

.99890  i  .06308 

.99801 

.08049 

.99676  23 

88 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894!  1.06337 

.99799 

.08078 

.9967'3  22 

39 

.01134 

.99994 

.02879 

.99959,  ,.04023 

.{li!S93  1.06366 

.99797 

.08107 

.99071 

21 

40 

.01164 

.99993 

.02908 

.99958  .04653 

.99892  .06395 

.99795 

.08136 

.99668  20 

41 

.01193 

.99993 

.02938 

.99957 

!  .04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.012-32 

.99993 

.02907 

.99956 

.04711 

.99889 

.00453 

.99792 

.08194 

.99001  IS 

43 

.01251 

.99992 

02996 

99955 

04740 

99888  .00482 

99790 

.08223 

99001  17 

44 

.01280 

.99992 

.03025 

.99954 

.04709 

.9C886  !  .06511 

.99788 

.08252 

.99059 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04796 

.  99885  i  .06540 

.99786 

.08281 

.99057 

15 

46 

.01.338 

.99991 

.03083 

.99952 

.04827 

.99883  .00:>0',l 

.99784 

.08310 

.99051 

14 

47 

.01307 

.99991 

.03112 

.99952 

.04856 

.9988211.00598 

.99782 

.08339 

.99052 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881  .00027 

.99780 

.08308 

.99019 

12 

49 

.01425 

.99990 

.0317'0 

.99950  .04914  .99879!  !  .00056 

.99778 

.08397 

.99047 

11 

50 

.01454 

.99989 

.03199 

.  99949  •  .  04943  _  .  99878  .  00085 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.  99948  '  .04972 

.99878 

.06714 

.99774 

-.08455 

.99042 

9 

52 

.01513 

.999S9 

.03257 

.9<H)  17  .05001 

.99875  i  .06743  .99772 

.08  4S4  ,.!>'.  Hi:  19 

8 

53 

-01542 

.99988 

.03286 

.99!  MO  .05030 

.9987311.06773 

.99770 

.08513  •.9116:37 

7 

51 

01571 

.99988 

.03316 

.99945  .06059 

.99872  1  .08802 

.99768 

.OS5I2  .99035 

6 

55 

.01600 

99987  ' 

.03345 

.99944  .05088 

.99870 

.06831 

.9<i7oo 

.08571 

.990:52 

6 

56 

.01629 

.99987 

.03374 

.9!  (913  .05117 

i  99809 

.06860  .99764 

.08(500 

99i',:;o 

4 

57 

.01658 

.99986 

.0:3403 

.99942;  .05146 

.99867 

.0(588!)  .99702 

.080-")  .990-J7 

3 

58 

.01687 

999SO 

.0:3432 

.99911  .05175 

.99866  .06918  .997(50 

.080.-8  .9i)(i-jr, 

2 

59 

.01716 

.99985 

.03461 

.99910  .05205 

.9980.1  .00917  .9975S 

.OH6H7  .99(522 

1 

60 

.(H715 

.W9S5 

.08490 

.99!«9  .05234 

.9980:!  .00970  .997'50 

.08710  .9%19 

0 

/ 

Cosin 

Sine 

Cosin 

Sine"  Cosin 

Sine 

Cosiu  ,  Sine 

Cosixi  Sine 

/ 

89° 

88°   1!   87° 

86° 

85° 

TABLE  X.— SINES  AND  COSINES. 


299 


5° 

6° 

7<>    j[    8« 

9° 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine  j  Cosin 

i 

~0  .08710 
1  .08745 

.99619 

.99617 

7l  0453  \99452 
.10482  .99449 

.12187 
.12216 

.99255 
.99251 

~  1391  7  .99027 
.13946  '.99023 

.15643 
.15672 

.98769 
.98764 

60 
59 

2  .08774 

.99614  .10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

58  ! 

3  !  .08803 

.9961211.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4  .08831 

.99609  .10569 

.99440 

.1-3302 

.99240 

.14033 

.99011 

.1575C 

.98751 

56 

5  .OSS60 

.9960? 

.10597 

.9943? 

.123311.99237  .14061 

.99006 

.1578? 

.98746 

55 

6  .08889 

.99604 

.10626 

.99431 

.12360  .91)2:33  .14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

99602 

.10655 

.99431 

.12:189  .99230 

.14119 

.98998 

.15845 

.98737 

53 

8 

.0894? 

.99599 

.10684 

.99423 

.12418  .99226 

.14148 

.98994 

.15873 

.98732 

52 

9 

.08976  .99596 

.10713  .99424 

.124471.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.  09005  i.  99594 

.10742  .99421 

.12475  .99219 

.14205 

.98986 

.15931 

.987'23 

50 

11 

.09034  .99591 

.10771  .99418 

.12504  .99215 

.14231 

.08982 

;.  15959 

.98718 

49 

12 

.09063  .99588 

.10800  .99415 

.12533  .99211 

.1426:1  1.98978 

.15988 

.98714 

48 

13 

.09092  .99.586 

.10S-*)  .99412 

.12562  .99208 

.14292 

.98973 

..16017 

.98709 

47 

14 

.09  121  '.99583 

.10858 

.99409 

.12591  .99204 

.14320 

.'.1896!) 

.16046 

.98704 

46 

15 

.09150  .9.I5SO 

.10887 

.99406 

.12620,.  99200 

.143-10 

.98966 

.16074 

.98700 

^15 

16  |  .09179 

.99578 

.109161.99402 

.126491.  99197 

.14378 

.88961 

.16103 

.98695 

44 

17  .09208 

.99575 

.10945  .99399 

.12678  .99193 

.1440? 

.9895? 

.16132 

.98690 

43 

18  .09237 

.99572 

.10973  |.  99396 

.127'06  .99189 

.14436 

i  98953 

.16160 

.98686 

42 

19  .09266 

.99570 

.11002  .99393 

.127.35  .99186 

.14464 

.9S948  .16189 

.98681 

41 

20  .09295 

.99567 

.11031  .99390 

.12764 

.9918:2 

.14493 

.98944  .16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940  .16246 

.98671 

39 

22  .09353 

.99562 

.11089!.  99383 

.12822 

.99175 

.14551 

.98936  .16275 

.9866? 

38 

23  .09382 

.99559 

.11118 

.99380 

.12851 

.99171 

.14580 

.98931  .16304 

.98662 

37 

24  I  .09411 

.99556 

.1114? 

.99377 

.12880  .99167, 

.14608 

.9892?  .163&3 

.98657' 

36 

25  '  .09440 

.99553 

.11176 

.99374 

.12908  '.99163 

.14637 

.98923  .16361 

.98652 

35 

26  .09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.98919;  .16390 

.98648 

34 

27  .09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.  98914  i  .1641!) 

.98643 

33 

28  .09527 

.99515 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910  .1644? 

.98638 

32 

29  .09556  .99542 

.11291 

.99300 

.1:3024 

.99148; 

.14752 

.HSJHHi  .16476 

.98633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.  14781  j.  98902 

.16505 

.98629 

30 

31 

.09614 

.9953? 

.11349 

.99351 

.13081 

.99141 

.14810 

.9889? 

.16533 

.98624 

29 

32  .09642 

.995:34 

.11378  .99351 

.13110 

.9913? 

.14838 

.98893  .16562 

.98619 

28 

33  !  .09671 

.99531 

.114071.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34  .09700 

.99528 

.11436 

.99344 

.13168 

.99129 

.14896 

.98884 

.16620 

.98609 

26 

35  .09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604 

25 

36  .09758 

.99523 

.11494 

.99337 

.13326 

.99122 

.14954 

.98876 

.16677 

.98600 

24 

37  .0978? 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595 

23 

38  .09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.9886? 

.16734 

.98590 

22 

39  .09845 

.99514 

.11580 

.9932? 

.13312 

.99110 

.15040 

.98863 

.16763 

.98585 

21 

40  1  .09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41  '  .09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

.98854 

1  .16820 

.98575  19 

42  .09932 

.995015 

.11667 

.99317 

.13399 

.99098 

.15126 

.98849 

.168-19 

.98570 

18 

43  .09901 

.99503 

.11696 

.99314 

.13427 

.99094 

.15155 

.98845 

:.  16878 

.98565 

17 

44  .09990 

.99500 

.11725 

.99310 

.1:3456  .99091 

.15184 

.98841 

.16906 

.98561!  16 

45  .10019 

.9949? 

.11754  -9930? 

.13485  .9908? 

.15212 

.98836 

!  .16935 

.98556 

15 

46 

.10048 

.99194 

.11783 

.99:303 

.13514  .99083 

.15241 

.98832 

.16964 

.98551 

14 

47 

.1007? 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135  .<)',)  is; 

.11869 

.99293 

.136001.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11898 

.99290 

.1  3629  !.  99067 

.15356 

.98814 

\  .17078 

.98531 

10 

51 

.10192 

.99479 

!  .11927 

.99286 

.13658  .99063 

.15385 

.98809 

.17107 

.98526 

9 

52 

.10221 

.99476 

.11956  .99283 

.1368?  .99059 

.15414 

.98805 

.17136 

.98681 

8 

53 

.10250 

.99473 

.119851.99279 

.13716  .99055 

.15442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.  1  201  4  j.  99276 

.13744  .99051 

.15471 

.98796 

.17193 

.9&511 

6 

55 

.10308 

.9946? 

.12043  .99272 

.13773  .99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

.1033^ 

.99464 

.12071  .99269 

.13802  .99043 

.15529 

.9878? 

.17250 

.98501 

4 

57 

10366 

.99461 

.12100  .99265 

.13831  .99039 

.15557 

.98782 

i  .17279 

.98496 

3 

58 

.103951.99458 

.  1  21  29  i  .  9926-2  .  1  3860  !  .  99035 

.15586 

.98778  .17308 

.98491 

2 

59 

.10424  .99455 

.  1  21  58  .  99258  .13889  .99031 

.15615 

98773  .17336 

.98486 

1 

60 

.10453  .99452 

.  1218?  j.  99255  .  .13917  .9902? 

.15643 

.98769 

,  .17365 

.98481 

0 

Cosin  |Sine7 

Cosin  j  Sine  l  Cosin  Sine 

Costa 

Sine  Cosin 

'Sine 

, 

84" 

83«   II   82» 

81° 

80° 

300 


TABLE  X.-SINES   AND   COSINES. 


10° 

11°       12° 

13°    ||    14° 

/ 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

SineJ  Cosin 

Sine  |  Cosin 

' 

0 

.17'365  .98481 

.19081 

.98  Ki.". 

!  20791  '.  97815 

.22495  .97437  .2  4  192  .97030 

60 

1 

.17393  .984761  .19109 

.98157 

.20820  .97809 

.  225:23  .  97430  ,  .  2422(  )  .  9702:'. 

59 

2 

.17422  .98471  .19138 

.98152 

.20848  .97803 

.22552  .97424  .2424., 

.97015 

,  58 

3 

.17451  .98466 

.19167 

.98146 

.20877  .97797 

.22580  .97417 

.24277 

.97008 

57 

4 

.17479  .98461 

.19195 

.98140 

.20905  .97791 

.22608  .97411 

.24305 

.97001 

!  56 

5 

.17508  .98455 

.19224 

.98135 

.  209:«  i.  97784 

.22637  .97404 

.24333 

.96994 

!  55 

6 

.17537  .98450 

.13252 

.98129 

.20962  .97778 

.  22665  i  4)7398 

.24362 

.9(1987 

54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.  96980 

53 

8  .17591  .98440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

.17623  .98135 

.19:338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24440 

.969(56 

51 

10 

.17651  .98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

150 

11  1.17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.228071.973(15 

.24503 

.96952 

49 

12  .17708 

.98420 

.1942.J 

.98096 

.21132 

.97743 

.22835  .9735S  .2-1531 

.96945 

i  48 

18 

.17737 

.98414 

.19452 

.98090 

.21161 

.97735 

.22863  .97861  .24559 

.969:37 

i  47 

14 

.17766 

.98109 

.19481 

.98084 

.21189 

.97729 

.22892  .97345H  .24587 

.969:30 

i  46 

15 

.  17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920  .97338  .2i01f, 

.!!6!)-23 

!  45 

16 

.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331!  .24644 

.96916 

44 

17 

.17852 

.98394; 

.19566 

.98067 

.21275 

.97711! 

.22977 

.97325,'  .2467:2 

.9(5909 

43 

18 

.17880 

.9S.JS9 

.19595 

.9,8061 

.21303 

.97705 

.28006 

.97318  .247'00 

.96902 

42 

19 

.17909  .98383 

.19623 

.98056 

.21331 

.97698 

.2:30*3  .97311 

.24728  .96894 

!  41 

20 

.17937 

.98378 

.19652 

.  98050 

.21360 

.97692 

.23062  .977304  .24750  .96887 

40 

21 

.17966 

.98373 

.19680 

.98044' 

.21388 

.97686 

.23090 

.97298 

.24784 

.9(1880!  39 

22 

.17995 

.98368 

.19709 

.98039 

.21417 

.97680 

.23118  .97291  |  .24813  .9687-! 

!  38 

23  .180281.98362 

.19737 

.98033 

.21445 

.97673 

.23146  .97284  !  .248411.96866 

37 

24  .1U0521.  98357 

.1976(5 

.98027 

.21474 

.97667 

.  23175  .  97278  '•  \  .  24869  1  .  96858 

36 

25  .18081 

.98332 

.19794 

.98021 

.21502 

.97661 

.23203 

.972711  .24897  .utW5l 

35 

26  .  18109 

.98347 

.19823 

98016; 

.215:30 

.97655  .23231 

.97264 

.24925 

.968-14 

34 

27  '.18138 

.98341 

.19851 

.  98010  i  .21559 

.97648 

.'23260 

.97257 

.24954  .96887 

33 

28 

.18166 

.98336 

.19880 

.98004 

.21587 

.97642 

.23288 

.97251 

.24982J.96829 

32 

29 

.  18195 

.98331 

.19908 

.97998  .21616 

.9763(5 

.23316 

.97244 

.250101.96822 

31 

30 

.18224 

.98325 

.19937 

.97992  .21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97987  .21072 

.97623 

.2a373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.19994 

.  97981  < 

.21701 

.97617, 

.23401 

.97223; 

.25094 

.9(5800 

28 

33 

.18309 

.98310 

.20022 

.97975  !  .21729 

.97611 

.23429 

.97217!  .25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

97969  .21758 

.97604 

.23458  .97210  .25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

97963 

.2178(5 

.97598 

.23486  .97203  !  .25179 

.96778 

25 

3(5 

.18395 

.98294 

.20108 

97958 

.21814 

.97592 

.23514 

.97196  .25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

97952 

.21843 

.97585 

.23542 

.97189  !  .25235 

.967(54 

23 

38 

.18452 

.98283 

.20165 

97916 

.21871 

.9757!) 

.23571 

.97182  .25:21)3  .9675<> 

22 

39 

.18481 

.98277! 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176  .252911.96749 

21 

40 

.18509 

.  98:272  i  .20222 

97934 

.21928 

.97566, 

.23627 

.97169  .25320  .9(5742 

20 

41 

.18538 

.98267  .20250 

97938 

.21956 

.97560'  .23656 

.97162  !  .25348 

.96734 

19 

42 

.18567 

.98261 

..  20279 

97922 

.21985 

.97553; 

.23684 

.97155  .25376 

.!«57'27 

18 

43 

.  18595 

.  98256  i!  .20307 

97916; 

.22013 

.97547! 

,23712 

.97148  .25404!.  96719 

17 

44 

.18624 

.98250  .20336 

97910 

.22041 

.97541 

.23740 

.97141  .254321.96712 

16 

45 

.18652  .98245  .20364 

97905 

.22070 

.97534 

.23769 

97K34  .25460  .9G705 

15 

46 

!  18681 

.98240  .20393 

.97899 

.22098  .97528 

.23797 

.97127  .  25488  L  96697 

14 

47 

.18710 

.98234  .20421 

97893 

.22126  .97521 

.23825 

.97120^  .25516  .96690 

13 

48 

.18738 

.98229 

.20450 

97887 

.22155 

.97515 

.23853 

.97113  .25545  .96682 

12 

49 

.18767 

.98223 

.20478 

.97881 

.22183  .97508 

.23882 

.97106  .25573  .96675 

11 

50 

.18795 

.98218  .20507 

.97875; 

.22212  .97502 

.23910 

.97100!  .25601  .96667 

10 

51 

.18824 

.98212  !  .20335 

.97869 

.22240 

.97496 

.23938 

.97093  .25629  .96660 

9 

52 

.188521.98207  j  .20563 

.978(53 

.2:2268  .97489  .23%<> 

.97086  .25657  .96653 

8 

53 

.18881  .98201  .20592  .97S5T  .22297  .97483  .23995 

.9707!)  .25685  .96645 

7 

54 

.18910  .981!)li  .20(520  .97851  .22325  .97476  .24023 

.97072  .25713  .90(1:58 

6 

55 

.189381.98190  .20649  .97845  .22353  .97470  .24051 

.9706.5  .25741  .96630 

5 

56 

.189071.9818.)  .20677,.  97839  .22382  .97463  .2407!)  .97058  .25769.96623 

4 

57 

.18995.98179  .20700.97833  .22410  .97457  .24108  .97051  .25798'  .96615 

3 

58 

.  1  9024  .  98  1  74  .  20734  .  978:27  .  22438  .  97450  .  24  1  36  i  .  97044  .  25826  .  96608 

2 

59 

.19052  98108  .207(13  .97821  .22467  .97444  .24164  .97037 

.25854  .96600 

1 

60 

.19081  .98163  .20791 

.97815  .22495 

.97437; 

.21192  .97030 

.25882  .96593 

0 

Cosin  Sine 

Cosin 

Sine 

Cosin 

l5ine 

Cosin  Sine 

Cosin  Sine 

t 

79° 

78° 

77° 

76° 

75° 

TABLE  X.— SINES   AND   COSINES. 


301 


15° 

1    16° 

17° 

18° 

19° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

T25882 

.96593  ".275»>i 

.%126  .29237 

.95630  730902 

.95106 

.32557 

.94552  60 

1  .25910 

.IMVoSo  .27592 

.96118  .2!  (265  .90622  .30929 

.95CJ7 

.32584 

.94542  59 

2  .25938 

.96578  .27620 

.  961  10  .  29293  .  9561  3  .  30957 

.95088 

.32612 

.94533-  58 

3  .2596(5 

.9(5570  .27648 

.96102  .29321  .95605  .30985 

.9507!) 

.32639 

.94523  57 

4  .25994 

.96562  .27676 

.96094  .29348  .95596  .31012  .95070 

.32667 

.  94514  1  56 

5  .20022 

.96555  .27704 

.96086  .29376 

.95588  .31040!.  95061 

.32694 

.94504!  55 

6  .2(5050 

.96547  .27731 

.  96078  .  29404  '  .  95579  .  31068 

.95052 

.32722 

.94495 

54 

7  .26079 

.96540'  .27759 

•96070  .29432 

.95571  .31095 

.95043 

.32749 

.94485 

53 

8  .2(5107 

.96532 

.27787 

.96062  .29460 

.95562 

.31123 

.9503:5 

.32777 

.94476 

52 

9  .2(5135 

.96524 

.27815 

.96054  .29487 

.95554  .31151 

.95024 

.32804 

.94466 

51 

10  .26103 

.96517 

.27843 

.96046  .29515 

.95545  .31178 

.95015 

.32832 

.94457 

50 

11  .26191 

.96509  .27871 

.96037  .29543 

.95536  .31  206  ;.  95006 

.32859  .94447 

49 

12  .26219 

.96502  .2;WJ'J 

.96029  .29571 

.95528  .31233  .94997 

.  32887  j.94438 

48 

13  .26247 

.96494  .27927 

.96021 

.29599 

.95519:  .31261  .«M9Ss 

.32914  .94428 

47 

14  .26275 

.96486 

.27955 

.96013  .29626 

.95511  .31289  .94979 

.32942  .94418 

46 

15  .26303 

.96479 

.27983 

.96006 

.  29654  .  95502  .  3131(5  .  94970 

.32969 

.94409 

45 

16  .26331 

.96471 

.28011 

.95997 

.29682  .95493  .31344  .94961 

.32997 

.94399 

44 

17  .26359 

.96463 

'.28039 

.  95989  1  .  29710  .  95485  1  .  31372  .  94952 

!  83024 

.94390 

43 

18  .26387 

.96456 

.28067 

.  95981  .  29737  .  95476  .  31399  1  .  94943 

.33051 

.94380 

42 

19  .2(5415 

.96448 

.28095 

.95972.  .29765 

.95467  .31427 

.94933 

.33079  .91370 

41 

20  .26443 

.96440 

.28123 

.95964 

.89798  .95459  .31454 

.94924 

.33106 

.94361 

40 

21  .26471 

.96433 

.28150 

.95956 

!.  29821  L  95450  .31482 

.94915 

.33134 

.94351 

39 

22  .26500 

.96425 

.28178 

.95948 

.29849  .95441  .31510 

.94906 

.33161 

.94342 

38 

23  .26528 

.96417 

.28206 

.95940 

.29876  '.95433  .31537 

.94897 

.33189 

.94332 

37 

24  .26556 

.96410 

.28234 

.95931 

.29904  .95424  .31565 

.94888 

.33216 

.94322 

36 

25  .26584 

.96402 

.28262 

.95923 

.29932  .95415  !  .31593 

.94878 

.33244 

.94313 

35 

26  .26612 

.96394 

.28290 

.95915 

.29960  .95407  .31620 

.94869 

.33271 

.94303 

34 

27  '  .26640 

.96386 

.28318 

.95907 

.29987  .95398  i  .31648 

.94860 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28346 

.95898 

.30015  .95389i!  .31675 

.94851 

.33326 

.9428! 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043  .953801  .31703 

.94842 

.88353 

.94274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372  .31730 

.94832 

.33381 

.94264 

30 

31 
32 

.26752 
.26780 

.96355 
.96347 

.28429 
.28457 

.95874 
.95865 

.30098  .95363  .31758 
.30126  .95354  .31786 

.94823 
.94814 

.3*3408 
.33436 

.94251 
.94245 

29 

28 

33  '  .26808 

.96340 

.28485 

.95857 

.301541.95345  .31813 

.94805 

-33463 

.94235 

27 

34  :  .26836  .96332 

.28513 

.95849 

.30182  .95337  .31841 

.94795 

;  .33490 

.94225 

26 

35  1.26864 

.96324 

.28541 

'.95841 

.30209 

.95328  .31868 

.94786 

.33518 

.94215 

25 

36 

.26MI2 

.96316 

.2S.-,6'J 

.95832 

.30237 

.95319  .31896 

.94777 

i  .33545 

.94206 

24 

37 

.26920 

:  96308 

.28597 

.95824 

.30265 

.95310 

.31923 

.94768 

i  .33573 

.94196 

23 

38 

.26918 

.93301 

.28625 

.95816 

.302921.95301 

.31951 

.94758 

.33600 

.94186  23 

39 

.26970 

.9:5293 

.28652 

.95807 

.30320  .95293 

.31979 

.94749 

.33627 

.94176  21 

40 

.27004  .  96285 

.28680 

.95799 

.30348  .95284 

.32006 

.94740 

.33655  .94167 

20 

41 

.27032  .96277  ! 

.28708 

.95791 

.30376  .95275  .32034 

.94730 

.33682 

.94157 

19 

42 

.27  060  '.90269 

.28736 

.95782 

.30403  .  95266  i  .32061 

.94721 

.33710  1.94147 

13 

43 

.27088 

.96261 

.28764 

.95774 

.30431  .95257  .32089 

.94712 

'  .33737  .94137 

17 

44 

.27116 

.96253 

.28792 

.95766 

.30459  .95248  .32116 

.947'02 

.337'64  .94127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30486  .95240 

.32144 

.94693 

i.  33792  j.  941  18 

15 

46 

.27172 

.96238 

'.  28847  1.95749 

.305141.95231 

.32171 

.91684 

.33819  .94108 

14 

47 

.27200 

.96230 

.28875 

.95740 

.30542  .9.V222 

.32199 

.94674 

.338461.94098 

13 

48 

.27228 

.96222 

.28903 

.95732  i  .305701.95213 

.32227 

.94665 

.33874|.94088 

12 

49 

.27256 

.96214 

.28931 

.95724 

.80597  .95204 

.32254 

.94656 

339011.94078 

11 

50 

.27284 

.96206 

.38959 

.95715 

.30625 

.95195 

.82282 

.946-ld 

.33929:.  94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.30653 

.95186 

.32309 

.94637 

.33956  .94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.95177 

.32337 

.94627 

.33983  .94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

.30708 

.95168 

.32364 

.94618 

.34011  .94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609 

.34038  .94029 

6 

55 

.27424 

.96166 

.29098 

.95673 

.30763 

.95150 

.32119 

.94599!  .34065  .94019 

5 

56 

.27452 

.96158 

.29126 

.95664 

.30791 

.95142  .32447 

.94590!  .34093  .94009 

4 

57 

.27480 

.96150 

.29154 

.9565(5  .30819 

.95133  !  .32474 

193580 

.34120  .93999 

3 

58 

.27508 

.96142 

.29182 

.95647!  .30846 

.95124  .32502 

.94571 

.34  147  1.93989 

2 

59 

.27536 

.96134 

.29209 

.95639  .30874 

.95115  .32529 

.94561 

.341  75!.  93979  1  1 

60 

.  27564  i.  96126 

.29237 

.95630 

.30902 

.95106  .32557 

.94552 

.34202 

.93969 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine" 

Cosiu 

"Sine" 

Cosin 

Sine" 

/ 

74° 

73° 

72° 

71° 

to° 

TABLE  X.-SINES  AND  COSINES. 


20°   ||   21° 

22° 

23°   i 

24°   ! 

' 

Sine  Cosin 

Sine 

Cosin 

Siae 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

~0 

.84202  .83989 

785837 

.  93358  !  737461 

.92718 

.39073 

.92050 

.40674 

'.91355 

fiO 

1 

.34229  .93959 

.35864 

.93348 

.37488 

.92707 

.39100 

.92039 

.40700 

.91343 

59 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92097 

.39127 

.93028 

.40737 

.91331 

58 

3 

.34284 

.93939 

.35918 

.9=3327 

.37'542 

.92080 

.39153 

.92010 

.40753 

.91319 

57 

4 

.34311  .93929 

.35945 

.93310 

.37509 

.92075 

.39180 

.92005: 

.40780 

.91307 

56 

5 

.3433!)  .93919 

.35973 

.93300 

.37595 

.92004 

.39207 

.91994; 

.40800 

.91295 

55 

6 

.34300  .93909 

.30000 

.93295 

.37022 

.92053 

.39234 

.91982 

.40833 

.1)1283 

54 

7 

.34393 

.93899 

.36027 

.93285 

.37049 

.93642 

.39200 

.911)71 

.40800 

.9JS72 

53 

8 

.34421 

.93389 

.30054 

.93274 

.37070 

.92031 

.39287 

.91959 

.40880 

.91200 

52 

9 

.34118 

.93879 

.30081 

.1CJ204  .37703 

.92020 

.39314 

.91948i 

.40913 

.91848 

51 

10 

.34475 

.93869 

.30108 

.'j:325:J 

.37730 

.92009 

.39341 

.  91930  j 

.40939 

.91230 

50 

11 

.34503 

.93859 

.36135 

.93243 

.37757 

.  92598  H.  39367 

.91925 

.409001.91224 

49 

12 

.84530 

.93849 

.36102 

.98382 

!  .37784 

.92587;  .39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93839 

.3or.«) 

.93222 

.37811 

.92570 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34584 

.93829 

.30217 

.93211 

.37838 

.39448 

.91891 

.41045 

.91188 

46 

15 

.34612 

.93819 

.30244 

.93201 

.37805  .92554 

.39-474 

.91879 

.41072 

.91176 

16 

.34639 

.93809 

.30271 

.93190 

.37892 

.92543  .39501 

.91808 

.41098 

.91104 

44 

17 

.34666 

.93799 

.30298 

.93180 

.37919 

.92532  .39528 

.91856 

.41125 

.91152 

43 

18 

.34694 

.93789 

.30325 

.93109 

.37940 

.92521  .39555 

.91845! 

.41151 

.1)1140 

42 

19 

.34721 

.93779 

.30352 

.93159 

.379?3i.  92510 

.39581 

.91833 

.41178 

91128 

41 

20 

.34748 

.99769 

.3637'9 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

.34775 

.93750 

.36406 

.93137 

.38026 

.92488 

.396.% 

.91810 

.41231 

.91104 

39 

22  .34803 

.93748 

.30434 

.93127 

.38053 

.92477 

.39001 

.91799  .41257 

.1)101)2 

38 

23  .34830 

.937:38! 

.30401 

.93110 

.38080 

.1)2406 

.39088 

.91787 

.41284 

.91080 

37 

24  1.34857 

.93728  .30488  .93100 

.38107 

.92-155 

.39715 

.91775 

.41310 

.91008 

36 

25  1.34884  .93718  .36515 

.93095 

.38134 

.C2444 

.39741 

.91704  .41337 

.91056 

35 

26 

.34912  .93708  .30542 

.9308-1  .38101 

.92432 

.39768 

.91752 

.41363 

.91044 

27  .34939 

.93098  .30509 

.9307'4j:  .38188 

.9242i 

.39795 

.91741 

.413901.91082  !53 

28  .3496*;  .93088  .30590 

.93063 

.38215 

.92410 

.39822 

.91729 

.41410 

.91020 

32 

29  .3400:5  -.03077  .30023 

.93052 

.::8241 

.1)2399  ..19K1S 

.91718 

.41443 

.111008 

31 

30 

.35021 

.93007 

.30050 

.93042 

.38208 

.92388  .39675 

.1.1700 

.41401)1.90990 

30 

31 

.35048 

.9305? 

.36677 

.93031 

.38295 

.02377  .  39902 

.91694 

.41190 

.90984 

29 

32  .35075  .93047  .30704 

.93020 

.36322 

.92300  .39928 

.91088 

.41522 

.90972 

28 

33  .351021.93037 

.30731 

.93010 

.38349 

.92355 

.39955 

.1)1071 

.41549 

'27 

34 

.351301.93020 

.30758 

.92999 

.38370 

.92343 

.39982 

.91GCO 

.41575 

!  1)01)48 

L'6 

35 

.35157  .93616 

.30785 

.92983  '  .38403 

.92:332 

.  .40008 

.91048 

.41602 

.5101)30 

25 

36 

.35184 

.93600 

.30812 

.92978  '  .384:30 

.92321 

:  .40035 

.91(W<5 

i  .41628 

.'.Ml'.,  -24 

24 

37 

.35211 

93590 

.308?]'.) 

.92907  .3S450 

.92310  .400021.91025 

.41  055  1.909  11 

23 

38 

.35239  .93585  |  .36867 

.92950  .38483 

.92299;!.  40088 

.91013!  .41081 

.!)08!)9 

22 

39 

.35230  .93575  i  .30894 

.92945 

.38510 

.922871  .40115 

.91C01 

.41707 

.90887 

21 

40 

.35293  .93505  .30921 

.92935  .385371.92276 

\  .40141 

.91590 

;  .41734 

.1)0875 

20 

41 

.35320 

.93555 

.36948 

.92921  .38504 

.92265 

.40168  .91578 

.41760 

.90803 

19 

42 

.35347 

.93514 

.36975 

.92913  !  .38591 

.1)2254  .4C195 

.91500  .41787 

.!K)851 

18 

43 

.35375  .93534 

.37002 

.92902  .38617 

.92243 

.40221 

.91555'  .il^Ti 

.90839 

17 

44 

.,35402;.93524 

.37029 

.92892 

.38044  .D22:]l 

.402-18 

.91543 

.41840 

.i)0820 

16 

45 

.35429;.  93514 

.37056 

.92881  .38671 

.92220  .4027'5 

.91531 

.41800 

.90814 

1C 

46 

.35450  .93503 

.37033 

.92870  .rjMKiS 

.92209  .40301 

.91519  .41892 

14 

47 

.354.-U  .1)3493 

.37110 

.92859  ;  .:587'25 

.921  9S  .40328 

.91508!  .41919 

!  907  90 

13 

48 

.35511  .93488 

.37137 

.92349:  .38752 

.92180 

.40355 

.JM490  .41945 

.90778 

12 

49 

.355:58  .9.'5  172 

.37104 

.92838  I  .38778 

.92175 

.40381 

.91484  .411)7':.' 

.90700 

11 

50 

.35505:.  93402 

.37191 

.92827  t  .38805 

.92104 

.40408 

.91472  .41998 

.1)0753 

10 

51 

.35592  '.93452 

.37218 

.92816  .38832 

.92152  !  .40434 

.91461 

I  .42024  .90741 

9 

52 

.85619  .98441 

.37245 

.92805   sss.7.) 

.!)::!  il  .40461 

.91449  .42051 

.90729 

8 

53 

.35647  .93431 

.37272 

.9271)4  ..'JSSSii 

.92180 

.40488 

.91437  ''  .42077 

.90717 

7 

54 

.35674 

.93120 

.37299 

.1)2784  '  .38912 

.92119 

.40514 

.91425-  .4210-1 

.90704 

6 

55 

.35701 

.93410 

.37320 

.92773  .38939 

.1)2107 

.40541 

.91411  .42130 

.9001)2 

5 

50 

.35728  .93400 

.37353 

.92762  .38966 

.92090  .40507 

.91402'  .42150 

U()i;m) 

4  ' 

57 

.35755  .93389 

.37380 

.92751  .3S!)l):5 

.92085  .40594 

.91390  .42183  .'JOOOH 

3 

58 

.35782  '.93379 

.37407 

.92740 

.39020 

.9207'.'5  .40(i21 

.91378  .42209 

.{XH5.W 

2 

59 

.35810  .93308 

.374:34 

.92729 

.39040 

.92062  .40647 

.<)i-:':00  .42235 

.90613 

1 

60 

.S5837  .9&S58 

.37401 

.1)2718 

.89073 

.92(150  .40674 

.91355 

.42262 

.IHXJ'51 

0 

t 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Siuc  i  Cosin 

Sine 

Cosin  j  Sine 

69° 

68* 

67°   II   66° 

65° 

VABLE   X.     SINES  AND  COSINES. 


303 


25° 

26° 

27° 

28° 

29° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  !  Cosin 

Sine  iCocin 

~6 

.42262 

.90631 

743837 

.89879 

745399 

.89101 

.469471.88295 

748481  !.H7'4(,  -2 

60 

i 

.42288 

.90618 

.43863 

.89867 

.45435 

.89087 

.469731.88281 

.48506 

.87418 

50 

2 

.42315 

.90606 

.43889 

.89854. 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

53 

3 

.42341 

.'J0594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45032 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.  44151  i.  89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45391 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.sr>:>H 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516  1.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542  .89532 

.4(5097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568  .89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.9)271 

.44594J.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.446201.89493 

.46175 

.88701 

.47716 

.8788.2 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646  '.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.446721.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698  .80454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724  '.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.46304  .88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.  46330  !.  88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146  .44854 

.89376 

.46407 

.88580 

.479461.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.4.3340 

.90120 

.44906  '.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43836 

.90108 

.44932  .89337 

.46484  .88539 

.48022!.  8771  5 

.49546 

.86863 

18  ! 

43 

.43392 

.90095 

.44958  .89324 

.465101.88526 

.  48048  1.87701 

.49571 

.86849 

17 

44  .43418 

.90082: 

.44984  .89311 

.46536 

.88512 

.48073  .87687 

.49596 

.86834 

16 

45  j.  43445 

.90070 

.45010  .89298 

.46561 

.88499 

.48099  .87673 

.49622 

.86820 

15  : 

46 

.43471 

.90057 

.45036  .89285 

.46587 

.88485 

.48124  .87659 

.49647 

.86805 

14  i 

47 

.43497 

.90045 

;  45062 

.89272 

.46613 

.88472 

.481501.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12  i 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007! 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86740 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981  i 

.45192 

.80-3f)(J 

,  .46742 

.88404 

.48277 

.87575 

.49798 

.8(5719 

8 

53 

.43654 

.89968 

.45218 

.81)193 

'  .46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89  IRQ 

'  ,46793 

.88377 

.4*328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.8836:} 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

i  .46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.80918; 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

>.W05 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

_0 

/ 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

"Sine 

/ 

64° 

63° 

62° 

61° 

60° 

3(H 


TABLE  X. --SINES   AND  COSINES. 


30*    I    31"   | 

32° 

33° 

at! 

y 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  !  Cosin 

Sine  !  Cosin 

f 

~0 

750000 

.80603 

.51504 

785717 

752992  ".84805  ~  54  404 

.8:3807  i  .55919  .82904 

00 

1 

.50025 

.80588 

.51529 

.85702 

.53017  .84789  .54488 

.88851 

.55943  .82887 

59 

2 

.50050 

.80573 

.51554 

.85087 

.530411.84774  .54513 

.88835 

.55908  .82871 

58 

S 

.50076 

.86559 

.51579 

.85072 

.53066  .84759  .54537 

.83819 

.55992 

.^855 

57 

4 

.50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.50010 

.S2S39 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115  .84728 

.54586 

.83788 

.50040 

.82*22 

55 

6 

.50151 

.86515 

.51653 

.85627 

.531  40  '.8471  2 

.54010 

.83772 

.56064 

.8;J800 

54 

7 

.50176 

.86501 

.51678 

.85612 

.53104 

.84697 

.54035 

.88766 

.56088 

.82790 

53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84081 

.54059 

.83740 

.56112 

.82773 

52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84600 

.54083 

.83724 

.56130 

.T.2757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.837^8  .50100 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84035' 

.54732 

.83692  .56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85530 

153288 

.84019 

.547'56 

.830701 

.5620IS 

.827'08 

48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.83000 

.56232 

.82092 

47 

14 

.5o;;r>-3 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82075 

46 

15 

.50377 

.80:584 

.51877 

.85491 

.53301 

.84573: 

.54829 

.83029 

.56280 

.82659 

45 

10 

.50403 

.80369 

.51902 

.85476 

.53386 

.84557! 

.54854 

.83613 

.56305 

.82043 

44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82020 

43 

18 

.50453 

.80340 

.51952 

.85446 

.5:3435 

.84526 

.541)02 

.88581 

.56353 

.82610 

42 

19 

.50478 

.80325 

.51977 

.85431 

.53460'  84511 

..'41)2? 

.83505 

.56377 

.82593 

41 

30 

.50503 

.86310 

.52002 

.8541G 

.534841.84495 

.54951 

.83549 

.50401 

.82577 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509  .84480  .54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

.80281 

.52051 

.85385 

.535341.84464 

.54999 

.83517 

.50449 

.82544 

38 

23 

.50578 

.86266 

.52076 

.85370 

.53553!.  84448 

.55024 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.535831.84433 

.55048 

.83485 

.56497 

.82511 

38 

25 

.50628 

.86237 

.52126 

.85340 

.530071.84417 

.550721.83469 

.50521 

.82495 

35 

26 

.50654 

.86222 

.52151 

.85835 

.536321.84402 

.55097 

.83453 

.50545 

.82478 

34 

27 

.50679 

.80207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.50509 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.50593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705  .84355 

.55169 

.83405 

.50017 

.82-429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730  .84339 

.551941.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754  .84324 

.55218 

.83373 

.56665 

.82396 

'29 

32 

.50804 

.86133 

.52299 

.85234 

.537791.84308 

.55242 

.83350 

.56689 

.82380 

,28 

33 

.50829 

.86119 

.52324 

.85218 

.5-3804 

.84292 

.55266  .83340 

.50713 

.:^:jG3 

27 

34 

.508.54 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83321 

.56730 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853  .84261 

.55315  .83308 

.5G7CO 

.KSJSO 

25 

36 

.50904 

.86074 

.52399 

.85173 

.538771.84245 

.55339 

.KfcJiW 

.5078-1 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.  53902  i.  84230 

.55363 

.83270 

.50808 

.82297' 

23 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.  55388  :  .83260  .56832 

.82:381 

22 

39 

.50979 

.80030 

.52473 

.85127! 

.53951 

.84193 

.55412 

.83244  .50850 

.82204 

21 

40 

.51004 

86015 

.52498 

.851  12  j 

.5397o 

.84182 

.55436  .83228  i  .56880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096' 

.54000 

.84167 

.55460 

.83212  '  .50904 

.82331 

19 

42 

.51054 

.85985 

.52547 

.85081 

.540241.84151 

.55484 

.83195  .50928!.  82214 

18 

43 

.51079 

.85970 

.52572 

.850661 

.64048  .84188 

.55509 

.83179  .50952  82198 

17 

44 

.51104 

.85950 

.52597 

.85051 

.54073  .84120 

.5.S533 

.83163  .509761.82181 

16 

45 

.51129 

.85941 

.52621 

.850351 

.54097 

.84104 

.55557 

.83147:  .57000  .82165 

15 

46 

.51154 

.85920 

.52646 

.85020 

.54122  .84088 

.55581 

.83131  .57024  .82148 

!  14 

47 

.51179 

.85911 

.53671 

.85005 

.54146  .84072 

.55605 

.83115  .57047 

.82132 

13 

48 

.51204 

.85896 

.52096 

.84989 

.54171 

.84057 

.55030 

.83098  .57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974; 

.54195 

.84041 

.55054 

.8:5082  .57095 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959; 

.54220  .84025 

.55078 

.83066  .57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84848 

.542441.84009 

.55702 

.&3050  .57143 

.82065 

•  9 

52 

.51304 

.858:?o 

.52794 

.84928, 

.512159  .83994 

.55726 

.83034  :  .57107 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017  i  .57191 

.82032 

;  7 

54 

.51354 

.85806 

.52844 

.84897 

.643171.88963 

.55775 

.83001  .57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.8-1882 

.543421.83946 

.55799 

.82985  .57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84800 

.54306  .83930 

.55823 

.82909  .57202 

.81982 

4 

57 

.51429 

.85702 

.52918 

.84851 

.54391  .83915 

.55847 

.82953  .57280 

.81'.)05 

3 

58 

.51454 

.85747 

.52943 

.84836 

.544151.83899  .55871 

.82930  .57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820: 

.54440  i  .83883!  .55895 

.82920  .57*34 

.81932 

1 

60 

.51504 

.85717 

.52992 

.  84805  : 

.54464-838671  .55919 

.82904  .57358 

.81915 

0 

/ 

Cosin  |  Sine 

Cosin 

Sine  !  Oosin  Sine  Cosin 

Sine 

j  Cosin 

Sine 

1 

59° 

58° 

57*       56° 

55° 

TABLE  X. -SINES   AND   COSINES. 


305 


35° 

36° 

37° 

38°    |    39° 

' 

Sine  SCosin 

Sine 

Cosin 

Sine 

Cosin  Sine  i  Cosin  j|  Sine 

Cosin 

/ 

~o 

.57358 

.81915 

.58779 

780902 

.00188 

.79864 

.61566  '78801!  .62932 

.77715  60 

1 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.615891.78783  .62955 

.77696  59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612  .78765  .62977 

.77678]  58 

3 

.57429 

.81865: 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660  57 

4 

.57453 

.818481 

.58873 

.808*3 

.60274 

.79793 

.61658 

.  78729  : 

.63022 

.77641 

56 

5 

.57477 

.81832  .58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55 

6 

.57501 

.  81815  i 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694! 

.63068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.S0782 

.60344 

.79741! 

.61726 

.78676 

.63090 

.77586 

53 

8 
9 

.57548 
.  57572 

.81782: 
.81765 

.58967].  80765 
.589901.80748 

.60367 
.60390 

.79723! 

.79706 

.61749 

.61772 

.78658 
.78640 

.63113 
.63135 

.77568 
.77550 

52 
51 

10 

.57590 

.81748 

.59014 

.80730 

.60414 

.  79688  ] 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713: 

.60437 

.  79671  !| 

.61818 

.78604! 

.63180 

.77513 

40 

12 

.57013 

.817141 

.59061 

.80696 

.60460 

.79653 

.61841  1.  78586  : 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.11864  ;.7B568 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.605061.79618 

.61887  i.78550 

.63248 

.77458  46  ; 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909  .78532 

.63271 

.77439  45 

16 

.57738 

.81647 

.59154 

.80627 

.60553 

.  79583  : 

.61982  .78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955  .78496 

.63316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61  978  ;.  78478 

.63338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.00622 

.79530  .62001  .78460 

.63361 

.77366 

41 

20 

.57833 

.81580, 

.59*48 

.80558 

.60645 

.79512  .62024  .78442  .63383 

.77347 

40 

21 

.57R57 

.81563 

.59272 

.80541 

.60668 

.79494!  .62046  .78424  !  .63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.  79477  i  .  62069  .  78405  .  63428 

.77310  38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459  .62092  .78387 

.63451 

.77292  37 

24 

.57928 

.81513! 

.59342 

.80489 

.60738 

.  79441  .  621  15  ,  .  78369  .  63473 

.77273  36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424  .62138.78351  .63496 

.77255  35 

26 
27 

.57976  .81479, 
.579991.81462' 

.59389  j.80455 
.594121.80438 

.60784 
.60807 

.794061  .621  60  '.  78333 
.79388'  .62183  .78315 

.63518 
.63540 

.77236  34 
.77218  33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371  .62206  .18297  .63563 

.77199  32 

29 

.580471.81428 

.  59459 

.80403 

60853 

.79353  .62229  .78279  .63585 

.77181 

31 

30 

.58070 

.81412: 

.59482 

.80386 

.60876 

.79335  .62251 

.78261;  .63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318  .62274 

.  78243  ! 

.63630 

.77144  29 

32 

.58118 

.  81378  : 

.59529 

.80351 

.60922 

.79300:  .62297  .18225 

.63653 

.77125!  28 

33 

.58141 

.81361! 

.59552 

.80334  .60945 

.79282  .62320  .18206 

.63675 

.77107  27 

34 

.58105 

.81344 

.595761.80316  .60968 

.79264  .62342  .78188 

.63698 

.770881  26 

35 

.58189 

.81327 

.59599 

.80299  .60991 

.79247  :  .  62365  L  78170 

.63720 

.  77070  j  25 

36 

.58212 

.81310:  .59622 

.80282  .61015 

.79229  .62388  .78152 

.63742 

.77051  24 

37 

.58236 

.812931  .59646 

.80264 

.61038 

.79211  .62411 

.78134 

.63765 

.77033  23 

38 

.58260 

.81276  .596691.80247 

.61061 

.79193'  .62433  .78116 

.63787 

.77014!  22 

39 

.58283 

.81259  .59693 

.80230 

.61084 

.79116  .62456;.  78098! 

.63810 

.76996  21 

40 

.58307 

.81242  .59716 

.80212 

.61107 

.79158  .624791.78079 

.63832 

.76977 

20 

41 

.56880 

.81225  .59739 

.80195'  .61130 

.79140 

.  62502  L  78061  : 

.63854 

.76959 

19 

42 

.58354 

.81208  i  .59763 

.80178 

.61153 

.79122 

.625241.78043 

.63877 

.76940 

18 

43 

.58378 

.81191!  .59786 

.80160  .61176 

.79105 

.62547 

.78025 

.63899 

.76921  17 

44 

.58401 

.81174' 

.59809 

.80143 

.61199 

.79C87 

.62570 

.78007 

.63922 

.76903  16 

45 

.584.25 

.81157' 

.59832 

.80125  .61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449 

.811401 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76866 

14 

47 

.58472 

.81123! 

.59879 

.80091! 

.61208 

.79033 

.62638^.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073  .61291 

.79016  !  .62660 

.77934 

.64011 

.76828  12 

49 

.58519 

.81089! 

.59926 

.80056  .61314 

.7S!««S  .62683 

.77916 

.64033 

.76810!  11 

50 

.58543 

.81072  .59949 

.80038 

.61337 

,7|B60  .62706 

.77897 

.64056 

.76191 

10 

51 

.58567 

.81055  1  .59972 

.80021  i  .61360 

.78962  .62728  .77879 

.64078 

.76772  9 

52 

.58590 

.81038:1  .59995 

80003  !  .61383 

.78944:  .62751!.  77861 

.64100 

..76154  8 

53 

.58614 

.81021  .60019 

.79986  i  .61406 

.  78926  .  62774  i  .  77843  .  641  23 

.76135;  7 

54 

.58037 

.  81004  !  .  60042  i  .  79968  .  61429 

.76908 

i  .62196 

.77824 

.64145 

.  79717  6 

55 

.58661 

.80987 

.60005 

.79951  .61451 

.7S891 

.62819 

.77806 

.64167 

.  76698 

5 

56 

.586841.80970' 

.60089 

i.  79934  .61474 

.78873  .62842 

.77788 

.64190 

.76679 

4 

57 

.58708 

.80953 

.60112 

.79916  .61497 

.78855  .6281)4 

.77769 

.64212 

.76661 

0 

58 

,  .58731 

.80936 

.60135 

t.  79899 

.  61520  i  .  78837  .  62887  !  .  77751 

.64234 

.76642 

2 

59 

.58755 

.80919 

.80158 

.79881 

.61543 

.78819  !  .62909  .777-33 

.64256 

.76623 

1 

60 

.  .58779 

.80902 

.60182 

:.  79864 

.61566 

.7SS01 

.62933 

.77715 

.64279 

.76604 

0 

/ 

Cosin  J"  Sine"  i 

Cosin  i  Sine 

Cosin 

Sine" 

Cosin 

Sine 

Cosin 

Sine 

/ 

54° 

53° 

52° 

51° 

50° 

306 


TABLE   X.— SINES   AND   COSINES. 


40° 

41° 

42° 

|    43° 

44° 

Sine 

i  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

' 

0 

7dl279 

.76604 

.65600 

.75471 

.60913 

".74314 

.Oo200 

.Wloo 

.094116  .71  '.131 

(JO 

1 

.64301 

.70586 

.656.28 

.75452 

.60935 

.74295 

.68221 

.73116 

.69487 

.71911 

59 

2 

.64323 

.76567 

.65650 

.75433 

.60950 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.04346 

.76548 

.6567'2 

.75414  .60978 

.74256 

.68204 

.7307-6 

.69529 

.71373 

1  57 

4 

.64368 

.76530 

.65694 

.7531)5  .60999 

.74237 

.682,35 

.73056 

.69549 

.71853 

56 

5 

.04390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68300 

.73036 

.69570 

.71833 

55 

6 

.61412 

.764'J2 

.65738 

.75350 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612  .71T92 

53 

8 

.64457 

.70455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72970 

.69633  .71772 

52 

9 

.64479 

.76438 

.65803 

.7521)9 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76398 

.65847 

.75201 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

.65359 

.75241 

.67172 

.74080 

.68455|.72897 

.69717 

.71691 

48 

13 

.64558 

.76361 

.65891 

.75222 

.67194 

.74061 

.68470  .72877 

.69737 

.71671 

47 

14 

.64590J.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.697'58 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

.646.35 

.76304 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76236: 

.65378 

.75146 

.67280 

.73983 

.68501 

72797 

.69821 

.71590 

43 

18 

.64679 

.76267 

.630  JO 

.75123 

.67301 

.73903 

.68582 

172777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.63022 

.75107 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229  .66044 

75088  .67344 

.7'3'J24 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

75089 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64788 

.76192 

.66038 

75050  i 

.67387 

.73885 

.68666 

.72097 

.699251.71488 

38 

23 

.64790 

.76173 

.68103 

75.030  ! 

.67409 

.73805 

.68688 

.72677 

.69946  .71468 

37 

24 

.64812 

.76154 

.66131 

75011 

.67430 

.73846 

.68709 

.72657 

.69966  j.  71447 

36 

25 

.64834 

.76135 

.66153 

74993 

.67452 

.73820 

.68730 

.72637 

.69987  .714271  35 

26 

.64856 

.76116 

.66175 

74973 

.67473 

.73808 

.68751 

.72617 

.700081.71407 

34 

27 
28 

.64878  .76097, 
.649011.76078 

.66197 
.66218 

74953 
74934 

.67495 
.67516 

.73787 
.73707 

.68772 
.68793 

.72597 
.  72577 

.70029  |.71386 
.70049  .71366 

33 

32 

29 

.64933 

.76059 

.66240 

74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30 

.64915 

.76041 

.66262 

74896 

.67559 

.73728 

.68835 

.  72537  | 

.70091 

.71325 

30 

31 

.61967 

.76022 

.68.284 

74876  ! 

.67580 

.73708 

.68857 

.  72517  ' 

.70112 

.71305 

29 

32 

.64989 

.76003 

.68308 

74857 

.67602 

.73683 

.68878 

.72497 

701321.71284 

28 

33 

.65011 

.75984 

.68327 

74838 

.67623 

.73669 

.68899 

.72477: 

.70153  .71264 

27 

34  '  .65033 

.75965  .66349 

74818 

.67645 

.73649 

.63920 

.72457; 

.70174  .71242 

26 

35  .65055 

.75946  .66371 

74799 

.67606 

.73623 

.68941 

.72437' 

.70195 

.71223 

35 

36 

.65077 

.75927 

.66393 

74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.65100' 

.75903 

.66414 

74760 

.67709 

.73590 

.68983 

.  72397 

.70836 

.71182 

23 

38  1  .65122 

.75883 

.60436 

74741 

.67730 

73570 

.69004 

.72377 

.70257 

.71162 

22 

39  .651441 

.75870 

.66458 

74722 

.67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.75832 

.68501 

746&3 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.63523 

74004 

.67816 

.73491 

.69088 

.72207 

.70339  .71  OSO 

18 

43 

.65232 

.75794 

.68543 

74644 

.67837 

.73472 

.69109 

.72277 

.70860  .71059 

17 

44 

.65254 

.75775 

.63586 

74625 

.678o9 

.73452 

.69130 

.72257 

.70381  .71039 

16 

45 

.65276 

.75756 

.60538 

74606 

.67880 

.73432 

.69151 

.72236 

.704011.7101!) 

15 

46 

.65298 

.75738 

.66610 

74586 

.67901 

.73413 

.69172 

.72216 

.70422  .70:>'38 

14 

47 

.65320 

.75719 

.666.321.74567;  .67923 

.73393 

.69193 

.7015)6 

.70443  .70978 

13 

48 

.65342 

.75700 

.666531.74548 

.67944 

.73373 

.69214 

.72176 

.70463  .705)57, 

12 

49 

.65364 

.75630 

.66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.70484  .70:137 

11 

50 

.65386 

.75661 

.66697 

.  74509  | 

.67987 

.73333 

.69256 

.72136 

.70505  .70316: 

10 

51 

.65408 

.75642 

.66718 

.74489  .68008 

.73314 

.69277 

.72116 

.70525  .70896' 

9 

52 

.65430 

.75623 

.66740  .74470  .68029 

.73294 

.69298 

.72095 

.705-10  .70875 

8 

53 

.65452 

.75604 

.66702  .74451  .68051 

.73274 

.69319 

.72075 

.705671.70855 

7 

54 

.65474 

!  75685 

.66783  .74431 

.68072 

.73254 

.69340 

.72055 

.70587  ,.70334 

6 

55 

.65496 

.75566 

.66805  .74412  .680931.73234 

.69361 

.72035 

.70608  -.70813 

5 

56 

.65518 

.75547 

.60827  .74392  .68115  .73215 

.69382 

.72015 

.70628..  70793 

4 

57 

.65.540 

.7.">2S  .0(5848  .74373 

.681361.73195 

.694031.71995 

.  70(349  .70772 

3 

58 

.65562 

.75509  .66870  .74858 

.68157 

.73175 

.694241.71974 

.70670  .70752 

2 

59 

.65584 

.75490 

.66891  .74*34 

.68179 

.73155 

.69445  .71!).-)  I 

.70690  .70731 

1 

60 

.65606  i.  75471 

.66913  .74314 

.68200 

.73135 

.60466 

.71934 

.70711  .70711  i 

0 

/ 

Cosin  Sine  Cosin  Sine  i  Cosin 

Sine 

Cosin 

Sine 

Cosin  i  Sine 

i 

49°       48° 

47° 

46°       45° 

XI.— NATURAL  SECANTS  AND  COSECANTS.        307 


/ 

SECANTS. 

f 

0 

1° 

2° 

30 

4° 

5° 

6° 

0 

1.00000 

1  00015 

1.00061 

1.00137 

1  .00244 

1.00382 

1  .00551 

60 

1 

00000 

00016 

00062 

00139 

00246 

00385 

00554 

59 

2 

00000 

00016 

00063 

00140 

00248 

00387 

oosr.r 

58 

3 

00000 

00017 

00004 

00142 

00250 

00390 

00560 

57 

4 

00000 

00017 

00065 

00143 

00253 

00392 

00563 

56 

5 

oooco 

00018 

00066 

00145 

00254 

00395 

00566 

65 

6 

00000 

00018 

00067 

00147 

00257 

00397 

00569 

54 

00000 

00019 

000(8 

00148 

00259 

00400 

00573 

63 

8 

00000 

00020 

00069 

00150 

002(51 

00403 

00576 

52 

9 

00000 

00020 

00070 

00151 

00263 

00405 

00579 

51 

10 

00000 

00021 

00072 

00153 

00265 

00408 

00582 

50 

11 

1.00001 

1.00021 

1.00073 

1.00155 

1.00267 

1.00411 

1.00585 

49 

12 

00001 

00022 

00074 

00156 

00269 

00413 

00588 

48 

18 

00001 

00023 

00075 

00158 

00271 

00416 

00592 

47 

14 

00001 

00023 

00076 

00159 

00274 

00419 

00595 

46 

15 

00001 

00024 

00077 

00161 

00276 

00421 

00598 

45 

16 

00001 

00024 

00078 

00163 

00278 

00424 

00601 

44 

17 

00001 

00025 

00079 

00164 

00280 

00427 

00604 

43 

18 

00001 

00026 

00081 

00166 

00282 

00429 

00608 

42 

19 

00002 

00026 

00082 

00168 

00284 

00432 

00611 

41 

20 

00002 

00027 

00083 

00169 

00287 

00435 

00614 

40 

21 

1.00002 

1.00028 

1.00084 

1.00171 

1.00289 

1.00438 

1.00617 

39 

22 

00002 

00028 

00085 

00173 

00:291 

00440 

00621 

38 

23 

00002 

00029 

00087 

00175 

00293 

00443 

00624 

37 

24 

00002 

00030 

00088 

00176 

00296 

00446 

00627 

36 

25 

00003 

00031 

00089 

00178 

00298 

00449 

00630 

35 

26 

00003 

00031 

00090 

00180 

00300 

00451 

00634 

34 

27 

00003 

00032 

00091 

00182 

00302 

00454 

00637 

33 

28 

00003 

00033 

00093 

00183 

00305 

00457 

00640 

32 

29 

00004 

00034 

00094 

00185 

00307 

00460 

00644 

31 

30 

00004 

00034 

00095 

00187 

00309 

00463 

00647 

30 

31 

1.00004 

1.00035 

1.00097 

1.00189 

1.00312 

1.00465 

1.00650 

29 

32 

00004 

00036 

00098 

001  DO 

00314 

00468 

00654 

28 

33 

00005 

00037 

00099 

00192 

00316 

00471 

00657 

27 

34 

00005 

00037 

00100 

00191 

00318 

00474 

00660 

26 

35 

00005 

00038 

00102 

001  D6 

00321 

00477 

00664 

25 

36 

00005 

00039 

00103 

00198 

00323 

00480 

00667 

24 

37 

00006 

00040 

00104 

00200 

00326 

00482 

00671 

23 

38 

00006 

00041 

00106 

00201 

00328 

00485 

00674 

22 

39 

00006 

00041 

00107 

00203 

00330 

00488 

00677 

21 

40 

00007 

00042 

00108 

00205 

00333 

00491 

00681 

20 

41 

1.00007 

1.00043 

1.00110 

1.00207 

1.00335 

1.00494 

1.00684 

19 

42 

00007 

00044 

00111 

00209 

00337 

00497 

00688 

18 

43 

00008 

00045 

00113 

00211 

00340 

00500 

00691 

17 

44 

00008 

00046 

00114 

00213 

00342 

00503 

00695 

16 

45 

00009 

00047 

00115 

00215 

00345 

00506 

00698 

15 

46 

00009 

00048 

00117 

00216 

00347 

00509 

00701 

14 

47 

00009 

00048 

00118 

00218 

00350 

00512 

00705 

13 

48 

00010 

00049 

00120 

00220 

00352 

00515 

00708 

12 

49 

00010 

00050 

00121 

00222 

00354 

00518 

00712 

11 

50 

00011 

00051 

00122 

00224 

00357 

00521 

00715 

10 

51 

1.00011 

1.00052 

1.00124 

1.00226 

1.00359 

1.00524 

1.00719 

9 

52 

00011 

00053 

00125 

00228 

00362 

00527 

00722 

8 

53 

00012 

00054 

00127 

00230 

00364 

00530 

00726 

7 

54 

00012 

00055 

00128 

00232 

00367 

00533 

00730 

6 

55 

00013 

00056 

00130 

00234 

00369 

00536 

00733 

5 

56 

00013 

00057 

00131 

00236 

00372 

00539 

00737 

4 

57 

00014 

00058 

00133 

00238 

00374 

00542 

007'40 

3 

58 

00014 

00059 

00!  34 

00240 

00377 

00545 

00744 

2 

59 

00015 

00060 

00136 

00242 

00379 

00548 

00747 

1 

60 

00015 

00061 

00137 

00244 

00382 

00551 

00751 

0 

f 

89° 

88° 

87° 

86° 

85° 

84° 

83° 

/ 

COSECANTS. 

308       XI.-NATURAL   SECANTS   AND  COSECANTS. 


/ 

SECANTS. 

] 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

0 

1.00751 

1.00983 

1.01247 

1.01543 

1.01872 

1.02234 

1.02630 

60 

1 

00755 

00987 

01251 

01548 

01877 

02240 

02637 

59 

2 

00758 

00991 

01256 

01553 

01883 

02247 

02(344 

58 

3 

00762 

00995 

01261 

01558 

01889 

02253 

02651 

57 

4 

007(35 

00999 

01265 

01564 

01895 

02259 

02658 

56 

5 

00769 

01004 

01270 

01569 

01901 

02266 

02665 

55 

6 

00773 

01008 

01275 

01574 

01906 

02272 

02672 

54 

7 

00776 

01012 

01279 

01579 

01912 

02279 

02679 

53 

8 

00780 

01016 

01284 

01585 

01918 

02285 

02686 

52 

9 

00784 

01020 

01289 

01590 

01924 

02291 

02693 

51 

10 

00787 

01024 

01294 

01595 

01930 

02298 

02700 

50 

11 

1.00791 

1.01029 

1.01298 

1.01601 

1.01936 

1.02:504 

1.02707 

49 

18 

00795 

01033 

01303 

01606 

01941 

02311 

02714 

48 

13 

.00799 

01037 

01308 

01611 

01947 

02317 

02721 

47 

14 

00802 

01041 

01313 

01616 

01953 

02323 

02728 

46 

15 

00806 

01046 

01318 

01622 

019r»9 

02330 

0273o 

45 

16 

00810 

01050 

01322 

01627 

01965 

02336 

02742 

44 

17 

00813 

01054 

01327 

01633 

01971 

02313 

02749 

43 

18 

00817 

01059 

01332 

01638 

01977 

02349 

02756 

42 

19 

00821 

01063 

01337 

01643 

01983 

02356 

02763 

41 

20 

00825 

01067 

01342 

01649 

01989 

02362 

02770 

40 

21 

1.00828 

1.01071 

1.01346 

1.01054 

1.01995 

1.02369 

1.02777 

39 

22 

00832 

01076 

01351 

01659 

02001 

02375 

02784 

'38 

23 

00836 

01080 

01356 

01665 

02007 

02382 

02791 

37 

24 

00840 

01084 

01361 

01670 

02013 

02388 

02799 

36 

25 

00844 

01089 

01366 

01676 

02019 

02395 

02806 

35 

26 

00818 

01093 

01371 

01681 

02025 

02402 

02813 

34 

27 

00851 

01097 

01376 

01687 

02031 

02408 

02820 

33 

28 

00855 

01102 

01381 

01692 

02037 

02415 

02827 

32 

29 

00859 

01106 

01386 

01698 

0-2043 

02421 

02834 

31 

30 

00863 

01111 

OJ391 

01703 

02049 

02428 

02812 

30 

31 

1.00867 

1.01115 

1.01895 

1.01709 

1.02055 

1.02435 

1.02849 

29 

32 

00871 

01119 

01400 

01714 

02061 

02441 

02856 

28 

33 

00875 

01124 

01405 

01720 

02067 

024  48 

028(J3 

27 

34 

00878 

01128 

OHIO 

01725 

02073 

02454 

02870 

26 

35 

00882 

01133 

01415 

01731 

02079 

02461 

02878 

25 

36 

00886 

01137 

01420 

01736 

(2085 

02468 

02885 

24 

37 

0;>890 

01142 

01425 

01742 

020P1 

02474 

02892 

23 

38 

00894 

01146 

01430 

01747 

02097 

02481 

02899 

22 

39 

00898 

01151 

01435 

01753 

02103 

0*488 

02907 

21 

40 

00902 

01155 

01440 

01758 

02110 

02494 

029  J  4 

20 

41 

1.00906 

1  01160 

1.01445 

1.01764 

1.02116 

1.02501 

1.02921 

19 

42 

00910 

01164 

01450 

01769 

02122 

02508 

02928 

18 

43 

00914 

01169 

01455 

01775 

02128 

02515 

02U36 

17 

44 

00918 

01173 

01461 

01781 

02134 

02521 

02913 

16 

45 

00922 

01178 

01466 

01786 

02140 

02528 

02950 

15 

46 

00920 

01182 

01471 

01792 

02146 

02535 

02958 

14 

47 

00930 

01187 

01476 

01798 

02158 

02542 

02965 

13 

48 

00934 

01191 

01481 

01803 

02159 

02548 

02972 

12 

49 

00938 

01196 

014.-6 

01809 

02165 

02555 

02980 

11 

50 

00942 

01200 

01491 

01815 

02171 

02562 

02987 

10 

51 

1.00946 

1.01205 

1.01490 

1.01820 

1.02178 

1.02569 

1  .02994 

9 

52 

00950 

01209 

01501 

01826 

02184 

02576 

03002 

8 

53 

00954 

0121' 

01506 

01832 

02190 

02582 

03009 

7 

54 

00958 

01219 

01512 

01837 

02196 

02589 

03017 

6 

55 

00962 

01223 

01517 

01843 

02203 

02596 

03024 

5 

56 

00966 

01228 

01522 

01849 

02209 

02603 

03032 

4 

57 

00970 

01233 

01527 

01854 

02215 

02610 

03039 

3 

58 

00975 

01237 

01532 

01860 

02221 

02617 

03046 

2 

59 

00979 

01242 

01537 

01866 

0222S 

02624 

03054 

1 

60 

00983 

01247 

01543 

01872 

02234 

02630 

03061 

0 

/ 

82° 

81° 

80° 

79° 

78° 

77° 

76° 

/ 

COSECANTS. 

XI.— NATURAL  SECANTS  AND  COSECANTS.       309 


SECANTS. 

, 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

0 

1.03061 

1.03528 

1.04030 

1.04569 

i.or.i  it; 

1.05762 

1.06418 

60 

1 

03069 

03536 

04039 

04578 

05156 

05773 

06429 

59 

2 

03076 

03544 

04047 

04588 

05106 

05783 

06440 

58 

S 

03084 

03552 

04056 

04597 

05176 

05794 

06452 

57 

4 

03091 

03560 

04065 

04006 

05186 

05805 

06463 

56 

5 

03099 

03568 

04073 

04616 

05196 

05815 

06474 

55 

6 

03106 

03576 

04082 

04625 

05206 

05826 

06486 

54 

7 

03114 

03584 

04091 

04635 

05216 

05836 

06497 

f.3 

8 

03121 

03592 

01100 

04041 

05226 

05847 

06508 

52 

9 

03129 

03601 

01108 

04653 

05236 

05858 

06520 

51 

10 

03137 

03609 

04117 

04663 

05246 

05869 

06531 

50 

11 

1.03144 

1.03617 

1.04126 

1.04672 

1.05256 

1  .05879 

1.06542 

49 

12 

03152 

03625 

04135 

04682 

0526<5 

05890 

06554 

4S 

13 

03159 

03633 

04144 

04691 

05276 

05901 

06565 

47 

14 

03167 

03642 

04152 

04700 

05286 

05911 

06577' 

46 

15 

03175 

03650 

04161 

04710 

05297 

05922 

065S8 

45 

16 

03182 

03658 

04170 

04719 

05307 

05933 

06600 

44 

17 

03190 

03666 

04179 

04729 

05317 

05944 

06011 

43 

18 

03198 

03674 

04188 

04738 

05327 

05955 

06622 

42 

10 

03-205 

03083 

04197 

04748 

05337 

05905 

06031 

41 

20 

03213 

03691 

04206 

04757 

05347 

05976 

06645 

40 

21 

1.03221 

1.03699 

1.04214 

1.04767 

1.05357 

1  .05987 

1  .06657 

39 

22 

03228 

03708 

04223 

04776 

05367 

05998 

06668 

38 

23 

03236 

03716 

04232 

04786 

05378 

06009 

06680 

37 

24 

03244 

03724 

04241 

04795 

05388 

06020 

06691 

36 

25 

03251 

03732 

04250 

04805 

05398 

06030 

06703 

35 

2G 

03259 

03741 

04259 

04815 

05408 

06041 

06715 

34 

'J7 

03-J67 

03749 

04268 

04824 

05418 

06052 

06726 

33 

28 

03475 

03758 

04277 

04834 

05429 

06063 

06738 

32 

29 

03282 

03766 

042S6 

04843 

05439 

06074 

06749 

31 

30 

03290 

03774 

04295 

04853 

05449 

06085 

06761 

30 

31 

1  .03298 

1.03783 

1.04304 

1.048G3 

1.05460 

1.06096 

1.06773 

29 

32 

03306 

03791 

04313 

04872 

05470 

06107 

06784 

28 

34 

03313 

03799 

04322 

04882 

05480 

06118 

06796 

27 

31 

03321 

03808 

04331 

04891 

05490 

06129 

06807' 

26 

35 

03329 

03816 

04340 

04901 

05501 

06140 

06819 

25 

36 

03337 

03825 

04349 

04911 

05511 

06151 

06831 

24 

37 

03345 

03833 

04358 

04920 

05521 

06162 

06843 

23 

38 

03353 

03842 

04367 

04930 

05532 

00173 

06854 

22 

39 

03360 

03850 

04376 

04940 

05542 

06184 

06866 

21 

40 

03368 

03858 

04385 

04950 

05552 

00195 

06878 

20 

41 

1.03376 

1.03867 

1.04394 

1.04959 

1.05563 

1.06206 

1.06889 

19 

42 

03384 

03875 

04403 

04969 

05573 

06217 

06901 

18 

43 

03392 

03884 

04413 

04979 

05584 

06228 

06913 

17 

44 

03400 

03892 

04422 

04989 

05594 

06239 

06925 

16 

45 

03408 

03901 

04431 

04998 

05604 

06250 

06936 

15 

46 

03416 

03909 

04440 

05008 

05615 

06261 

06948 

14 

47 

03424 

03918 

04449 

05018 

05625 

06272 

00960 

13 

48 

03432 

03927 

04458 

0502H 

05636 

06283 

0097'2 

12 

49 

03489 

03935 

04408 

05u38 

05646 

06295 

06984 

11 

50 

03447 

03944 

04477 

05047 

05657 

06306 

06995 

10 

51 

1.03455 

1.03952 

1.04486 

1  .05057 

1.05667 

1.06317 

1.07007 

9 

52 

03468 

03961 

04495 

05067 

05678 

06328 

07019 

8 

53 

03471 

03969 

04504 

05077 

05088 

06339 

07031 

54 

03479 

03978 

04514 

05087 

05699 

06350 

07043 

6 

55 

03487 

03987 

04523 

05097 

05709 

06362 

07055 

5 

56 

03495 

03995 

04532 

05107 

05720 

06373 

07067 

4 

57 

03503 

04004 

04541 

05116 

05730 

06384 

07079 

3 

58 

03512 

04013 

04551 

05126 

05741 

00395 

07091 

2 

59 

03520 

04021 

04560 

0513(5 

05751 

00407 

07103 

1 

60 

03528 

04030 

04569 

05146 

05762 

06418 

07115 

0 

/ 

75° 

74° 

73° 

7--)0 

71° 

70° 

69° 

COSECANTS. 

*>10       XI.— NATUKAL  SECANTS   AND  COSECANTS. 


t 

SECANTS. 

21° 

22° 

23° 

24° 

25° 

20° 

27° 

0 

1.07115 

1.07853 

1.08636 

1.09464  1 

.  10388 

1.11260 

1.12233 

eo 

1 

071  -.'6 

07866 

08649 

09478 

10353 

11276 

12249 

59 

2 

07138 

07879 

08663 

09492 

10368 

11292 

12266 

58 

3 

07150 

07892 

08676 

09506 

10383 

11308 

12283 

57 

4 

07162 

07904 

08690 

09520 

10398 

113-23 

12299 

56 

5 

07174 

07917 

08703 

09535 

10413 

11339 

12316 

55 

6 

07186 

07930 

08717 

09549 

10428 

11355 

12333 

54 

f 

07199 

07943 

08730 

09563 

10443 

11371 

12349 

53 

8 

0;211 

07955 

08744 

09577 

10458 

11387 

12366 

52 

9 

07228 

07968 

087T>7 

09592 

10-173 

11403 

12383 

51 

10 

07235 

07981 

08771 

09606 

10488 

11419 

12400 

50 

11 

1.07247 

1  07994 

1.08784 

1.09620  1 

.10503 

1.11435 

1.12416 

49 

12 

07259 

08006 

08798 

09635 

10518 

11451 

12433 

48 

13 

07-71 

08019 

0881  1 

09649 

10533 

11467 

12450 

47 

14 

•  07283 

08032 

08825 

09663 

10549 

11483 

12467 

46 

15 

07295 

03045 

08839 

09678 

10564 

11499 

12484 

45 

16 

07307 

08058 

08852 

09692 

10579 

11515 

12501 

44 

17 

07320 

08071 

08866 

09707 

10594 

li:31 

12518 

43 

18 

07332 

08084 

08880 

09721 

10609 

11547 

12534 

42 

19 

073  14 

08097 

08893 

09735 

1002.-) 

11563 

12551 

41 

20 

09856 

08109 

08907 

09750 

10640 

11579 

12568 

40 

21 

1.07368 

1.08122 

1.08921 

1.09764  1 

.10655 

1.11595 

1.12585 

S9 

22 

07380 

08135 

08934 

09779 

10G70 

11611 

12602 

38 

23 

07393 

08148 

08948 

09793 

10086 

11627 

12619 

37 

24 

07405 

08101 

08962 

09808 

10701 

11643 

12638 

36 

25 

07417 

08174 

08975 

09822 

10716 

11659 

12653 

35 

26 

07429 

08187 

08989 

09837 

10731 

11675 

12070 

34 

27 

07442 

03200 

09003 

09851 

10747 

11691 

12687 

33 

28 

07454 

08213 

09017 

098C6 

10762 

11708 

12704 

32 

29 

07166 

08226 

09030 

09880 

10777 

11724 

12721 

31 

30 

07479 

08239 

09044 

09895 

10793 

11740 

12738 

30 

31 

1.07491 

1.08252 

1.09058 

1.09909  1 

.10808 

1.11756 

1.12755 

29 

32 

07503 

082(55 

09072 

09924 

10824 

11772 

12772 

28 

88 

07516 

08278 

09086 

09939 

10839 

11789 

12789 

27 

34 

07528 

08291 

09099 

09953 

10854 

11805 

12807 

26 

35 

07540 

08305 

09113 

09968 

10870 

11821 

128C4 

25 

36 

07553 

08318 

09127 

09982 

10885 

11838 

12841 

24 

37 

07505 

08331 

09141 

09997 

10901 

1  1  854 

12858 

23 

38 

07578 

08344 

09155 

10012 

10916 

11870 

12875 

22 

39 

07590 

OH357 

09169 

10026 

10932 

11886 

12892 

21 

40 

07602 

08370 

09183 

10041 

10947 

11903 

12910 

20 

41 

1.07615 

1.08383 

1  09197 

1.10055  1 

.10963 

1.11919 

1.12927 

19 

42 

07627 

08397 

09211 

10071 

10978 

11936 

12944 

IS 

43 

07640 

08410 

09224 

10085 

10994 

11953 

12961 

17 

44 

07852 

08423 

09238 

10100 

11009 

11908 

12979 

16 

45 

07665 

08436 

09252 

10115 

11025 

11985 

12998 

15 

46 

07677 

08449 

09266 

10130 

11041 

12001 

13013 

14 

47 

07690 

08463 

09280 

10144 

11056 

12018 

13031 

13 

48 

07702 

08476 

09294 

10159 

11072 

12034 

13048 

12 

49 

07715 

08489 

09308 

10174 

11087 

12051 

13065 

11 

50 

07727 

08503 

09323 

10189 

11103 

12067 

13083 

10 

51 

1.07740 

1.08516 

1  .09337 

1.10204  1 

11119 

1.12084 

1.13100 

9 

52 

07752 

08529 

09351 

10218 

11134 

12100 

13117 

8 

53 

07765 

08542 

09365 

10233 

11150 

12117 

13135 

7 

54 

07778 

08556 

09379 

10248 

11166 

12133 

13152 

6 

55 

07790 

08569 

09393 

10263 

11181 

12150 

13170 

5 

56 

07803 

08582 

09407 

10278 

11197 

12166 

13187 

4 

57 

07816 

08596 

09421 

10293 

11213 

12183 

13205 

3 

58 

07828 

08609 

09435 

10308 

11229 

12199 

13222 

2 

59 

07841 

OSG23 

OR449 

10323 

11214 

12216 

13240 

1 

60 

07853 

08636 

09464 

10338 

11260 

12233 

13-J57 

0 

/ 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

/ 

COSECANTS. 

XI.  — NATURAL  SECANTS  AND  COSECANTS. 


SECANTS.                    j   , 

28° 

29° 

30° 

31° 

32° 

33° 

34° 

0 

1.1  3-257 

1.14335 

1.15470 

1.16663  1. 

17918 

1.19236 

1.20622 

60 

1 

13275 

14354 

15489 

16684 

17939 

19259 

2064f 

59 

2 

13292 

14372 

15509 

16704 

17961 

19281 

20669 

58 

3 

13310 

14391 

15528 

16725 

17982 

19304 

20693 

57 

4 

133-27 

14409 

15548 

16745 

18004 

19327 

20717 

r.e 

13345 

14428 

15567 

16766 

18025 

19349 

20740 

55 

I 

13362 

14446 

15587 

16786 

18047 

19372 

20764 

54 

7 

13380 

14465 

15IJ06 

16806 

18068 

19394 

20788 

53 

8 

13398 

14483 

1562(3 

16827 

18090 

19417 

20812 

52 

9 

13415 

1  4502 

15645 

16848 

18111 

19440 

20836 

51 

10 

13433 

14521 

15665 

16868 

181*3 

19463 

20859 

50 

11 

1.13451 

1.14539 

1.15684 

1.16889  1. 

18155 

1.19485 

1.20883 

49 

12 

13468 

14558 

15704 

16909 

18176 

19508 

20907 

48 

13 

13480 

14576 

15724 

16930 

18198 

19531 

20931 

47 

14 

13504 

14595 

15743 

16950 

18220 

19554 

20955 

46 

15 

13521 

14614 

15763 

16971 

18241 

19576 

20979 

45 

16 

13539 

14632 

15782 

16992 

18263 

19599 

21003 

44 

17 

13557 

14651 

15802 

17012 

18285 

19622 

21027 

43 

18 

13575 

11670 

15822 

17033 

18307 

19645 

21051 

42 

19 

13593 

14689 

15841 

17054 

1S32S 

19668 

21075 

41 

20 

13610 

14707 

15861 

17075 

18350 

19691 

21099 

40 

21 

1.13628 

1.14726 

1.15881 

1.17095  1 

18372 

1.19713 

1.21123 

39 

22 

13646 

14745 

15901 

17116 

18394 

19736 

21147 

38 

23 

13664 

14764 

15920 

17137 

18416 

19759 

21171 

37 

24 

13682 

14782 

15940 

17158 

18437 

19782 

21195 

36 

25 

13700 

14801 

15960 

17178 

18459 

19805 

21220 

35 

20 

13718 

14820 

15980 

17199 

18481 

19828 

21244 

34 

27 

13735 

14839 

16000 

17220 

18503 

19851 

21268 

33 

28 

13753 

14858 

16019 

17241 

18525 

19874 

21292 

32 

29 

13771 

14877 

16039 

17262 

18547 

19897 

21316 

31 

30 

13789 

14896 

16059. 

17283 

18569 

19920 

21341 

30 

31 

1.13807 

1.14914 

1.16079 

1.17304  1. 

18591 

\.19944 

1.21365 

29 

32 

13825 

14933 

16099 

17325 

18613 

19967 

21389 

28 

38 

13843 

14952 

16119 

17346 

18635 

19990 

21414 

27 

34 

13861 

14971 

16139 

17367 

18657 

20013 

21438 

26 

35 

13879 

14990 

16159 

17388 

1867'9 

20036 

21462 

25 

36 

13897 

15009 

16179 

17409 

18701 

20059 

21487 

24 

37 

13916 

15028 

16199 

17430 

18723 

20083 

21511 

23 

38 

13884 

15047 

16219 

17451 

18745 

20106 

21535 

22 

39 

13952 

15066 

16239 

17472 

18767 

20129 

21560 

21 

40 

13970 

15085 

16259 

17493 

18790 

20152 

21584 

20 

41 

1.13988 

1.15105 

1.16279 

1.17514  1. 

18812 

1.20176 

1.21609 

19 

42 

14006 

15124 

16299 

17535 

18834 

20199 

21633 

18 

43 

14024 

15143 

16319 

17566 

18856 

20222 

21658 

17 

44 

14042 

15162 

16339 

17577 

18878 

20246 

21682 

16 

45 

14061 

15181 

16359 

17598 

18901 

20269 

217'07 

15 

46 

14079 

15200 

16i80 

17620 

1SD2.-J 

20292 

21731 

14 

47 

14097 

15219 

16400 

17641 

1S945 

20316 

21756 

13 

48 

14115 

15239 

16420 

17662 

181)67 

20339 

21781 

12 

4!) 

14134 

15258 

16440 

17683 

18990 

20363 

21805 

11 

50 

14152 

15277 

16460 

17704 

19012 

20386 

21830 

10 

51 

1.14170 

1.15296 

1.16481 

1.17726  1.19034 

1.20410 

1.21855 

9 

52 

14188 

15315 

16501 

17747 

19057 

20433 

21879 

8 

53 

14207 

15335 

16521 

17768 

19079 

20457 

21904 

7 

54 

14225 

15354 

16541 

17790 

19102 

20480 

21929 

6 

55 

14243 

15373 

16562 

17811 

19124 

20504 

21953 

5 

5(5 

14262 

15393 

165S2 

17832 

19146 

20527 

2  1978 

4 

57 

142SO 

15412 

16602 

17854 

191(19 

20551 

22003 

3 

58 

14299 

15431 

16623 

17875 

19191 

20575 

22028 

2 

59 

14317 

15451 

16643 

1789U 

19214 

20598 

22058 

1 

60 

14335 

15470 

16663 

17918 

11)236 

20622 

.22077 

0 

Sl° 

60° 

5!>° 

68° 

Sf» 

66° 

55° 

COSECANTS. 

: 

312       XI.—  NATURAL  SECANTS  AND  COSECANTS. 


, 

SECANTS. 

t 

35° 

36° 

37° 

38° 

39° 

40° 

41° 

0 

1.22077 

1  .23607 

1.25214 

1.26902  1.28676 

1.30541 

1.32501 

60 

1 

22102 

23633 

25241 

26931 

28706 

30573 

32535 

59 

2 

22127 

23659 

25269 

26960 

28737 

30605 

32568 

58 

3 

22152 

23685 

25296 

26988 

28767 

30636 

32602 

57 

4 

22177 

23711 

25324 

27017 

28797 

30668 

32636 

56 

5 

22202 

23738 

25351 

27046 

288->8 

30700 

32669 

55 

6 

22227 

23764 

25379 

27075 

28858 

30732 

327'03 

54 

7 

22252 

23790 

25406 

27101 

28889 

30764 

32737 

53 

8 

22277 

23816 

25434 

27133 

28919 

30796 

32770 

52 

9 

22302 

23843 

25462 

27162 

28950 

30829 

32804 

51 

10 

22327 

23869 

25489 

27191 

28980 

30861 

32838 

50 

11 

1.22352 

1.23895 

1.25517 

1.27221  1 

.29011 

1.30893 

1.32872 

49 

12 

22377 

23922 

25545 

27250 

29042 

30925 

32905 

48 

13 

22402 

23948 

25572 

27279 

29072 

30957 

32939 

47 

14 

22428 

23975 

25600 

27308 

29103 

30989 

32973 

46 

15 

22453 

24001 

25628 

27337 

291:53 

31022 

33007 

45 

16 

22478 

24028 

25656 

27366 

29164 

31054 

33041 

44  ' 

ir 

22503 

24054 

25683 

27396 

29195 

31086 

33075 

43 

18 

22528 

24081 

25711 

27425 

29226 

31119 

33109 

42 

19 

22554 

24107 

25739 

27454 

29256 

31151 

33143 

41 

20 

22579 

24134 

25767 

27483 

29287 

31183 

33177 

40 

21 

1.22604 

1.24160 

1.25795 

1.27513  1 

.29318 

1.31216 

1.33211 

39 

22 

22629 

24187 

25823 

27542 

29349 

31248 

33245 

38 

23 

22655 

24213 

25851 

27572 

29380 

31281 

33279 

37 

24 

22680 

24240 

25S79 

27601 

29411 

31313 

33314 

36 

25 

22706 

24267 

25007 

27630 

29442 

31346 

33348 

35 

26 

22731 

24293 

25935 

27660 

29473 

31378 

33382 

34 

27 

22756 

24320 

25963 

27689 

29504 

31411 

33416 

33 

28 

22782 

24347 

25991 

27719 

29535 

31443 

33451 

32 

29 

22807 

24373 

26019 

27748 

29566 

31476 

33485 

31 

30 

22833 

24400 

26047 

27778 

29597 

31509 

33519 

30 

31 

1.22858 

1.24427 

1.26075 

1.27807  1 

.29628 

1.31541 

1.33554 

29 

32 

22884 

24454 

26104 

27837 

29659 

31574 

33588 

28 

33 

22909 

24481 

26132 

27867 

29(590 

31607 

33622 

27 

34 

2-2935 

24508 

26160 

27896 

29721 

31640 

33657 

26 

35 

22960 

24534 

26188 

27926 

29752 

31672 

33691 

25 

36 

22986 

24561 

26216 

27956 

29784 

31705 

33726 

24 

37 

23012 

24588 

26245 

27985 

29815 

31738 

33760 

23 

38 

23037 

24615 

26273 

28015 

29846 

31771 

33795 

22 

39 

23063 

24642 

26301 

28045 

29877 

31804 

33830 

21 

40 

23089 

24669 

26330 

28075 

29909 

31837 

33864 

20 

41 

1.23114 

1.24696 

1.26358 

1.28105  1.29940 

1.31870 

1.33899 

19 

42 

23140 

24723 

26387 

28134 

29971 

31903 

33934 

18 

43 

23166 

24750 

26415 

28164 

30003 

31936 

33968 

17 

44 

23192 

24777 

26443 

28194 

30034 

31969 

34003 

16 

45 

23217 

24804 

26472 

28224 

30066 

32002 

34038 

15 

46 

23243 

24832 

26500 

28254 

30097 

32035 

34073 

14 

47 

23269 

24859 

26529 

28284 

30129 

32068 

34108 

13 

48 

23295 

24886 

26557 

28314 

30160 

32101 

34142 

12 

49 

23321 

24913 

26586 

28344 

30192 

32134 

34177 

11 

50 

23347 

24940 

26615 

28374 

30223 

32168 

34212 

10 

51 

1  .-23373 

1.24967 

1.26643 

1.28404  1.30255 

1.32201 

1.34247 

9 

52 

23399 

24995 

26672 

28434 

302S7 

32234 

34282 

8 

53 

23424 

25022 

26701 

28464 

30318 

32267 

34317 

7 

54 

23450 

25049 

26729 

28495 

30350 

32301 

34352 

6 

55 

23476 

25077 

26758 

28525 

30382 

32334 

34387 

5 

56 

23502 

25104 

26787 

28555 

30413 

32368 

34423 

4 

57 

23529 

25181 

26815 

28585 

30445 

32401 

34458 

3 

58 

23555 

25159 

26844 

28615 

30477 

32434 

34493 

2 

59 

23581 

25186 

26873 

28646 

30509 

32468 

34528 

1 

60 

23607 

25214 

26902 

28676 

30541 

32501 

34563 

0 

54° 

63° 

52° 

r>i° 

50° 

49° 

48° 

/ 

COSECANTS. 

XL— NATURAL   SECANTS   AND   COSECANTS.       313 


SECANTS. 

; 

42° 

43° 

440 

45° 

46° 

470 

48° 

0 

1.34563 

1.36733 

1.39016 

1.41421 

1.43956 

1.46628 

1.49448 

60 

1 

34599 

36770 

39055 

41463 

43999 

46674 

49496 

59 

2 

34634 

36807 

39095 

41504 

44042 

46719 

49544 

58 

3 

34669 

36844 

39134 

41545 

44086 

46765 

49593 

57 

4 

34704 

36881 

39173 

41586 

44129 

46811 

49641 

56 

5 

34740 

36919 

39212 

41627 

44173 

46857 

49690 

55 

6 

3477'S 

3C956 

39251 

41669 

44217 

46903 

49738 

54 

7 

34811 

36993 

39291 

41710 

44260 

46949 

49787 

53 

8 

34846 

37030 

39330 

41752 

44304 

46995 

'  49835 

52 

9 

34882 

37'068 

39369 

41793 

44347 

47041 

49884 

51 

10 

34917 

37105 

39409 

41835 

44391 

47087 

49933 

50 

11 

1.34953 

1.37143 

1.39448 

1.41876 

1.44435 

1.47134 

1.49981 

49 

12 

34988 

37180 

39487 

41918 

44479 

47180 

50030 

48 

13 

35024 

37218 

39527 

41959 

44523 

47226 

50079 

47 

14 

35060 

37255 

39566 

42001 

44567 

47272 

50128 

46 

15 

35095 

37293 

39606 

42042 

44610 

47319 

50177 

45 

16 

35131 

37330 

39646 

42084 

44654 

47365 

50226 

44 

17 

35167 

37368 

39685 

42126 

44698 

47411 

50275 

43 

18 

35203 

37406 

39725 

42168 

44742 

47458 

50324 

42 

19 

35238 

37443 

39764 

4-J210 

44787 

47504 

50373 

41 

20 

35274 

37481 

39804 

42251 

44831 

47551 

50422 

40 

21 

1.35310 

1.37519 

1.39844 

1.42293 

1.44875 

1.47598 

1.50471 

39 

22 

35346 

87556 

39884 

42335 

44919 

47644 

50521 

38 

23 

35382 

37594 

39924 

42377 

44963 

47691 

50570 

37 

24 

35418 

37632 

39963 

42419 

45007 

47738 

50619 

36 

25 

35454 

37670 

40003 

42461 

45052 

47784 

50669 

35 

26 

35490 

37708 

40043 

42503 

45096 

47831 

50718 

34 

27 

35526 

37746 

40083 

42545 

45141 

47878 

50767 

33 

28 

35562 

37784 

40123 

42587 

45185 

47925 

50817 

32 

29 

35598 

37822 

40163 

42630 

45209 

47972 

50866 

31 

30 

35634 

37860 

40203 

42672 

45274 

48019 

50916 

30 

31 

1.35670 

1.37898 

1.40243 

1.42714 

1.45319 

1.48066 

1.50966 

29 

32 

35707 

37936 

40283 

42756 

45363 

48113 

51015 

28 

33 

35743 

37974 

40324 

42799 

45408 

48160 

51065 

27 

34 

35779 

38012 

40364 

42841 

45452 

48207 

51115 

26 

35 

35815 

38051 

40404 

42883 

45497 

48254 

51165 

25 

36 

35852 

38089 

40444 

42926 

45542 

48301 

51215 

24 

37 

35888 

38127 

40485 

42968 

45587 

48349 

51265 

23 

38 

35924 

38165 

40525 

4301  1 

45631 

48396 

51314 

22 

39 

35961 

38204 

40565 

43053 

45676 

48443 

51364 

21 

40 

35997 

38242 

40606 

43096 

45721 

48491 

51415 

20 

41 

1.36034 

1.38280 

1.40646 

1.43139 

1.45766 

'.48538 

1.51465 

19 

42 

36070 

38319 

40687 

43181 

45811 

48586 

51515 

18 

43 

36107 

38357 

40727 

43224 

45856 

48633 

51565 

17 

44 

36143 

38396 

40768 

43267 

45901 

48681 

51615 

16 

45 

36180 

38434 

40808 

43310 

45946 

48728 

51665 

15 

46 

36217 

38473 

40849 

43352 

45992 

48776 

51716 

14 

47 

36253 

38512 

40890 

43395 

46C37 

48824 

51766 

13 

48 

36290 

38550 

40930 

43438 

46082 

48871 

51817 

12 

49 

36327 

38589 

40971 

43(81 

46127 

48919 

51867 

11 

50 

36363 

38628 

41012 

43524 

4617-3 

48967 

51918 

10 

51 

1.36400 

1.38666 

1.41053 

1.43567 

1.46218 

3.49015 

1.51968  * 

9 

52 

36437 

38705 

41093 

43610 

46263 

49063 

52019 

8 

53 

36474 

3S744 

41134 

43653 

46309 

49111 

52069 

54 

36511 

38783 

41175 

43696 

40354 

49159 

52120 

6 

55 

36548 

38822 

41M6 

43739 

46400 

49207 

52171 

5 

56 

30585 

38860 

41257 

43783 

46445 

49255 

52222 

4 

57 

36602 

38899 

41298 

43826 

46491 

49303 

52273 

3 

58 

36G.M) 

38938 

41339 

43869 

46537 

49351 

52323 

2 

59 

30696 

38977 

41380 

43912 

46582 

49399 

52374 

1 

60 

86788 

39016 

41421 

43966 

40028 

494  4  « 

52425 

0 

,   1    «• 

46° 

45° 

44" 

43° 

42  o 

41° 

f 

I 

COSECANTS. 

314       XI.— NATURAL  SECANTS   AND  COSECANTS. 


t 

SECANTS. 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

0 

1.52425 

1.55572 

1.58902 

1.62427  1 

.66164 

1.70130 

1.7  13  15 

60 

1 

52476 

55626 

58959 

62487 

66228 

70198 

74417 

59 

2 

52527 

55080 

59016 

62548 

66292 

70267 

74490 

5S 

3 

52579 

55734 

59073 

62609 

66357 

70335 

74562 

57 

4 

52630 

55789 

59130 

62669 

66421 

70403 

74035 

56 

5 

52681 

55843 

59188 

62730 

60180 

70472 

74708    55 

6 

52732 

55897 

59245 

62791 

60550 

70540 

74781    54 

7 

52784 

55951 

59302 

62852 

00015 

70609 

74851 

T.3 

8 

52835 

56005 

59360 

62913 

66679 

70677 

74927 

52 

<) 

52886 

56060 

59418 

62974 

00711 

70746 

75000 

51 

10 

52938 

56114 

59475 

63035 

66809 

70815 

75073    50 

11 

1.52989 

1.56169 

1.59533 

1.63096  1 

.66873 

1.70884 

1.75146 

49 

12 

63041 

56223 

59590 

63157 

WG93S 

70953 

75219 

48 

13 

53092 

56278 

59048 

63218 

67003 

710-22 

75293 

47 

14 

53144 

56332 

59706 

63279 

67068 

71091 

75366 

40 

15 

53196 

56387 

59764 

63841 

67133 

71160 

75440 

45 

16 

53247 

56442 

59822 

63402 

67199 

71229 

75513 

41 

17 

53299 

56497 

59880 

63464 

67204 

71298 

75587 

43 

18 

53351 

56551 

59938 

63525 

67329 

71368 

75661 

42 

19 

53403 

56606 

59996 

63587 

67394 

71437 

75734 

41 

20 

53455 

56661 

60054 

63648 

67460 

71506 

75808 

40 

21 

1  .53507 

1.56716 

1.60112 

1.63710  1.67525 

1.71576 

1.75882 

39 

22 

53559 

56771 

60171 

63772 

67591 

71646 

75956 

38 

23 

53611 

56826 

60229 

63834 

0705G 

71715 

76031 

37 

24 

53603 

56881 

60287 

6:5895 

67722 

71785 

76105 

36 

25 

53715 

56937 

60346 

63957 

67788 

71855 

76179 

35 

26 

53768 

56992 

60404 

64019 

67853 

71925 

76253 

34 

27 

53820 

57047 

60463 

64081 

67919 

71995 

76328 

S3 

28 

53872 

57103 

60521 

64144 

67985 

72065 

76402 

32 

29 

53924 

57158 

60580 

64206 

68051 

72135 

76477 

31 

30 

53977 

57213 

60639 

64268 

68117 

72205 

76552 

30 

31 

1.54029 

1.57209 

1.60698 

1.64330  1.68183 

1.72275 

1.76C26 

29 

32 

54082 

57324 

60756 

64393 

68250 

72346 

76701 

28 

33 

54134 

57380 

60815 

64455 

68316 

72416 

76776 

27 

34 

54187 

57436 

60874 

045  18 

68382 

72487 

76851 

26 

35 

51240 

57491 

60933 

64580 

68449 

72557 

70920 

36 

54292 

57547 

60992 

01043 

68515 

72628 

77001 

24 

37 

54345 

57603 

61051 

64705 

68582 

72698 

77077 

23 

38 

54398 

57659 

61111 

64T6.S 

68C48 

72769 

77152 

22 

39 

54451 

57715 

61170 

6483  1 

68715 

72810 

77227 

21 

40 

54504 

57771 

61229 

64894 

68782 

72911 

77303 

20 

41 

1.54557 

1.57827 

1.612S8 

1.64957  1.68848 

1.72982 

1.77378 

19 

42 

54610 

57883 

61348 

G5020 

68915 

73053 

77454 

18 

43 

54663 

57939 

61407 

65083 

68982 

73124 

77530 

17 

44 

54716 

57995 

61467 

65146 

G9049 

73195 

77606 

10 

45 

54769 

58051 

61526 

65209 

69116 

73267 

7768J 

15 

46 

54822 

58108 

61586 

65272 

69183 

73338 

77757 

14 

47 

54876 

58164 

61646 

65336 

69250 

73409 

77833 

13 

48 

54929 

58221 

61705 

65399 

69318 

73481 

77910 

12 

49 

54982 

58277 

61765 

65462 

69385 

73552 

77986 

11 

50 

55036 

58333 

61825 

65526 

69452 

73624 

78062 

10 

51 

1.55089 

1  .58390 

1.61885 

1.65589  1.69520 

1.73696 

1.78138 

9 

52 

55143 

58447 

61945 

65653 

6JI5S7 

73768 

78215 

8 

53 

55196 

58503 

62005 

65717 

69655 

73840 

78291 

7 

54 

55250 

58500 

62065 

657PO 

69723 

73911 

78368 

6 

55 

55303 

58617 

62125 

C.5S4  1 

69790 

73983 

7S445 

5 

56 

55357 

58674 

62185 

65908 

69MS 

74056 

78521 

4 

57 

55411 

58731 

62240 

65972 

69920 

74128 

78598 

3 

58 

55465 

58788 

62306 

66036 

69994 

74200 

78675 

2 

59 

55518 

58845 

62366 

60100 

70002 

74272 

78752 

1 

60 

55572 

5S902 

62427 

60164 

70130 

74315 

78829 

0 

e 

40° 

39° 

38° 

37° 

3«° 

35° 

34° 

f 

COSECANTS. 

XI.— NATURAL  SECANTS  AND  COSECANTS.         315 


/ 

SECANTS. 

/ 

56° 

57° 

58° 

59° 

60° 

61° 

62° 

0 

1.78829 

1.83608 

1.88708 

1.94160 

2.00000 

2.06267 

2  13005 

60 

1 

789C6 

83690 

88796 

94254 

00101 

06375 

13122 

59 

2 

78984 

83773 

88884 

94349 

00202 

06483 

13239 

58 

3 

79061 

83855 

88972 

94443 

00303 

06592 

13356 

57 

4 

79138 

83938 

89060 

94537 

00404 

06701 

K3473 

56 

5 

79216 

84020 

89148 

94632 

00505 

06809 

13590 

55 

6 

79293 

84103 

89237 

94726 

00607 

06918 

13707 

54 

79371 

84186 

89325 

94821 

00708 

07027 

13825 

53 

8 

794-19 

842G9 

89414 

94916 

00810 

07137 

13942 

52 

9 

79527 

84352 

89503 

0501  1 

00912 

07246 

14000 

51 

10 

79604 

84435 

89591 

95106 

01014 

07356 

14178 

50 

11 

1.796S2 

1.84518 

1.89680 

1.95201 

2.01116 

2.07465 

2.14296 

49 

I-,' 

79761 

84601 

89769 

95296 

01218 

07575 

14414 

48 

13 

79839 

84685 

851838 

95392 

01320 

07685 

14533 

47 

14 

79917 

84768 

89948 

95487 

01422 

07795 

14651 

46 

15 

79995 

84852 

90037 

95583 

01525 

07905 

14770 

45 

16 

80074 

84935 

90126 

95678 

01628 

08015 

14889 

44 

17 

80152 

85019 

90216 

95774 

01730 

08126 

15008 

43 

18 

80231 

85103 

90305 

95870 

01833 

OS-J36 

15127 

42 

19 

80309 

85187 

90395 

95966 

01936 

08347 

15246 

41 

20 

80388 

85271 

90485 

96002 

02039 

08458 

15366 

40 

81 

1.80467 

1.85355 

1.90575 

1.96158 

2.02143 

2  08569 

2.15485 

39 

22 

80546 

85439 

90665 

96255 

02246 

08680 

15005 

38 

23 

80625 

85523 

90755 

96351 

02349 

08791 

15725 

37 

24 

80704 

85608 

90845 

96448 

02453 

08903 

15845 

36 

25 

80783 

85692 

90935 

96544 

02557 

09514 

15965 

35 

26 

80862 

85777 

91026 

96641 

02661 

09126 

1C085 

34 

27 

80942 

85861 

91116 

96738 

02765 

09238 

16206 

33 

28 

81021 

85946 

91207 

96835 

02869 

09350 

16326 

32 

29 

81101 

86031 

91297 

96932 

02973 

09462 

16447 

31 

30 

81180 

86116 

91388 

97029 

03077 

09574 

16568 

30 

31 

1.81260 

1.86201 

1.91479 

1.97127 

2.03182 

2.09686 

2.16689 

29 

32 

81340 

86286 

91570 

97224 

03286 

09799 

16810 

28 

S3 

81419 

815371 

91661 

97322 

03391 

09911 

16932 

27 

34 

81499 

86457 

91752 

974'JO 

03496 

10024 

7053 

26 

35 

81579 

86542 

91844 

97517 

03601 

10137 

7175 

25 

36 

81G59 

86627 

91935 

97615 

03706 

10250 

7297 

24 

37 

81740 

86713 

92027 

97713 

03811 

10363 

7419 

23 

38 

81820 

86799 

92118 

97811 

03916 

10477 

7541 

22 

39 

81900 

86885 

92210 

97910 

04022 

10590 

7663 

21 

40 

81981 

86990 

92302 

98008 

04128 

10704 

17786 

20 

41 

1.820G1 

1.87056 

1.92394 

1.98107 

2.04233 

2.10817 

2.17909 

19 

42 

82142 

87142 

92486 

98205 

04339 

10931 

18031 

18 

43 

82222 

87229 

92578 

9&304 

04145 

11045 

18154 

17 

44 

82303 

87315 

92670 

98403 

04551 

11159 

18277 

16 

45 

82384 

87401 

92762 

98502 

04658 

11274 

18401 

15 

46 

82465 

87488 

92855 

98601 

04764 

11388 

18524 

14 

47 

82546 

87574 

92947 

98700 

04870 

11503 

18648 

13 

48 

82627 

87661 

93040 

98799 

04977 

11617 

18772 

12 

.49 

82709 

87748 

93133 

98899 

05084 

11732 

18895 

11 

50 

82790 

87834 

93226 

98998 

05191 

11847 

19019 

10 

51 

1.82871 

1.87921 

1.93319 

1.99098 

2.05298 

2.11983 

2.19144 

9 

52 

82953 

88008 

93412 

99198 

05405 

12078 

19268 

8 

53 

83034 

88095 

93505 

99298 

05512 

12193 

19393 

7 

54 

83116 

88183 

93598 

99398 

05619 

12309 

19517 

6 

55 

83198 

83270 

93692 

99498 

05727 

12425 

19642 

5 

56 

83280 

88357 

93785 

99598 

G6S35 

12540 

19767 

4 

57 

83362 

88445 

93879 

99698 

05942 

12657 

19892 

3 

58 

83444 

88532 

93973 

99799 

06050 

12773 

20018 

2 

59 

835-26 

88620 

940*56 

99899 

06158 

12889 

20143 

1 

00 

83608 

88708 

94160 

2.00000 

06267 

13005 

20269 

0 

t 

33° 

32° 

31° 

30° 

29° 

28° 

27° 

/ 

COSECANTS. 

316       XI.— NATURAL   SECANTS  AND  COSECANTS. 


SECANTS. 

63° 

«4° 

05° 

«6° 

67° 

OS0 

<•,!> 

0 

2.20269 

2.28117 

2.36620 

2.45859 

2.55930 

2.66947 

2.79043 

60 

1 

20395 

28558 

36768 

46020 

56106 

67139 

79254 

59 

2 

20521 

28390 

36910 

46181 

56282 

67332 

79466 

58 

3 

20647 

28526 

37064 

46342 

56458 

67525 

79679 

57 

4 

20773 

28063 

37212 

46504 

56634 

67718 

79891 

56 

5 

20900 

28800 

37361 

46665 

56811 

67911 

80104 

55 

6 

21026 

28937 

37509 

46W27 

56988 

68105 

80318 

51 

7 

21153 

29074 

37658 

46989 

57165 

68299 

80531 

53 

8 

21280 

29211 

37808 

47152 

57342 

68494 

80746 

5\! 

9 

21407 

29349 

37957 

47314 

575JO 

68689 

80960 

51 

10 

21535 

29487 

38107 

47477 

57698 

68884 

81175 

50 

11 

2.21662 

2.29625 

2.38256 

2.47640 

2.57876 

2.69079 

2.81390 

49 

12 

21790 

29763 

38406 

47804 

58054 

69275 

81605 

48 

13 

21918 

29901 

38556 

47967 

58*13 

69471 

81821 

47 

14 

2:2045 

30040 

38707 

48131 

58412 

69667 

82037 

46 

15 

22174 

30179 

388:>7 

48295 

58591 

69864 

82254 

45 

16 

22302 

30318 

39008 

484.-)'.) 

58771 

70061 

82471 

44 

17 

22430 

30457 

39159 

48624 

58950 

70258 

82688 

43 

18 

22559 

30596 

39311 

48789 

59130 

70455 

82906 

42 

19 

22688 

30735 

39462 

48954 

59311 

70653 

83124 

41 

20 

22817 

30875 

39614 

49119 

59491 

70851 

83342 

40 

21 

2.22946 

2.31015 

2.39766 

2.49284 

2.59672 

2.71050 

2.83561 

39 

22 

23075 

31155 

39918 

49450 

59853 

71249 

83780 

38 

23 

23205 

31295 

40070 

49616 

60035 

71448 

83999 

37 

24 

23334 

31436 

40222 

49782 

60217 

71647 

84219 

36 

25 

23464 

31576 

40375 

49948 

60399 

71847 

84439 

35 

26 

23591 

31717 

40528 

50115 

60581 

72047 

84659 

34 

27 

23724 

31858 

40681 

50282 

60763 

72247 

84S80 

33 

28 

23855 

31999 

40835 

50449 

60946 

72448 

85102 

32 

29 

23985 

32140 

40988 

50617 

61129 

72649 

85323 

31 

30 

24116 

32282 

41142 

50784 

61313 

72850 

85545 

30 

31 

2.24247 

2.32424 

2.41296 

2.50952 

2.61496 

2.73052 

2.85767 

29 

32 

24378 

32566 

41450 

51120 

61680 

73254 

85990 

28 

88 

24509 

32708 

41605 

51289 

61864 

73456 

86213 

27 

34 

24640 

32850 

41760 

51457 

62049 

53659 

86437 

26 

35 

24772 

32993 

41914 

51626 

62234 

73862 

86661 

25 

36 

24903 

33135 

4207'0 

51795 

62419 

74065 

86885 

24 

37 

25035 

33278 

42225 

51965 

62604 

74269 

87109 

23 

38 

25167 

33422 

42380 

52134 

62790 

74473 

87334 

22 

39 

25300 

33565 

42536 

52304 

62976 

74677 

87560 

21 

40 

25432 

33708 

42692 

52474 

63162 

74881 

87785 

20 

41 

2.25565 

2.33852 

2.42848 

2.52645 

2.63348 

2.75086 

2.88011 

19 

42 

25697 

33996 

43005 

52815 

63535 

75292 

88238 

18 

43 

25830 

34140 

43162 

52986 

63722 

75497 

88465 

17 

44 

25963 

34284 

43318 

53157 

63909 

75703 

88692 

16 

45 

26097 

34429 

43476 

53329 

64097 

75909 

88920 

15 

46 

26230 

34573 

48633 

53500 

64285 

76116 

89148 

14 

47 

26364 

34718 

43790 

53672 

64473 

76323 

89376 

13 

48 

26498 

34863 

43948 

53845 

64662 

76530 

89(505 

12 

49 

26632 

35009 

44106 

54017 

64851 

76737 

898:34 

11 

50 

26766 

35154 

44264 

54190 

65040 

76945 

90063 

10 

51 

2.26900 

2.35300 

2.44423 

2  54363 

2.65229 

2.77154 

2.9025)3 

9 

52 

27035 

35446 

44582 

54536 

65419 

773(12 

90524 

8 

53 

2?169 

35592 

44741 

54709 

65609 

77571 

90754 

7 

54 

27304 

35738 

44900 

54883 

65799 

77780 

90986 

6 

55 

27439 

35885 

45059 

55057 

65989 

77990 

91217 

5 

56 

27574 

36031 

45219 

55231 

66180 

78200 

91449 

4 

57 

27710 

36178 

45378 

55405 

66371 

78410 

91681 

3 

58 

27845 

36325 

45539 

55580 

66563 

78621 

91914 

2 

59 

27981 

36  473 

45699 

55755 

66755 

7H83-.J 

92147 

1 

60 

28117 

36620 

45859 

55980 

66947 

79043 

92380 

0 

26° 

25° 

24° 

23° 

"2  "2° 

21° 

20° 

/ 

COSECANTS. 

XI.— NATURAL   SECANTS    AND   COSECANTS.        31' 


/ 

g 

EC  AN'! 

S. 

/ 

70° 

71° 

-00 

73° 

740 

75° 

76° 

0 

2.9-2380 

3.07155 

8.88607 

3.42030 

3.62796 

3.86370 

4.13357 

60 

1 

92614 

07415 

238!>7 

42356 

63164 

86790 

13839 

59 

a 

92849 

07075 

24187 

42683 

63533 

87211 

14323 

58 

3 

93083 

07936 

24478 

43010 

63903 

876:33 

14809 

57 

4 

93318 

08197 

24770 

43337 

64274 

88056 

15295 

56 

5 

93554 

08459 

25062 

43666 

64645 

88179 

15782 

55 

6 

93790 

087:21 

25355 

43995 

65018 

88904 

16271 

54 

7 

94026 

08983 

25648 

44324 

65391 

89330 

16761 

53 

8 

94263 

09246 

25942 

44655 

65765 

89756 

1725-2 

52 

9 

94500 

09510 

26-237 

449H6 

66140 

90184 

17744 

51 

10 

94737 

09774 

26531 

45317 

66515 

90613 

1S238 

50 

11 

2.94975 

3.10038 

3.268-27 

3.45650 

3.66892 

3.91042 

4.18733 

49 

12 

95213 

10303 

27123 

45983 

67269 

91473 

19228 

43 

13 

95452 

10568 

27420 

46316 

67647 

91904 

19725 

47 

14 

95091 

10834 

27717 

40651 

68025 

92337 

202-24 

46 

15 

9593  J 

11101 

28015 

46986 

68405 

92770 

20723 

45 

16 

96171 

11867 

28313 

47321 

68785 

93204 

2V2-24 

44 

17 

96411 

11635 

28612 

47658 

69167 

93640 

217^6 

43 

18 

96652 

11903 

28912 

47995 

69549 

94076 

22229 

42 

19 

96  93 

12171 

2921-2 

48333 

69931 

94514 

227:^4 

41 

20 

97135 

12440 

29512 

48671 

70315 

94952 

23-239 

40 

21 

2.97377 

3.12709 

3.29814 

3.49010 

3.70700 

3.95392 

4.23746 

39 

22 

97619 

12979 

30115 

49350 

71085 

95832 

24-255 

38 

23 

97862 

13249 

30418 

49691 

71471 

96274 

24764 

37 

24 

98106 

13520 

307-21 

50032 

71858 

96716 

25275 

36 

25 

98349 

13791 

31024 

50374 

72246 

97160 

25787 

35 

26 

98594 

14063 

313-28 

50716 

72635 

97604 

26300 

34 

27 

98838 

14335 

31633 

51060 

73024 

98050 

26814 

33 

28 

99083 

14608 

31939 

51404 

73414 

98497 

27330 

32 

29 

99329 

14881 

3-2244 

51748 

73806 

98944 

27847 

31 

30 

99574 

15155 

32551 

52094 

74198 

99393 

28366 

30 

31 

2.99821 

3.15429 

3.3-2858 

3.52440 

3.74591 

3.99843 

4.28885 

29 

32 

3.00067 

15704 

33166 

52787 

74984 

4.00293 

29406 

28 

33 

00315 

15979 

33474 

53134 

75379 

00745 

29929 

27 

34 

00562 

16255 

33783 

53482 

75775 

01198 

30452 

26 

35 

00810 

16531 

34092 

53831 

76171 

01652 

30977 

25 

30 

01059 

16808 

34403 

54181 

76568 

02107 

31503 

24 

3V 

01308 

17085 

34713 

54531 

76966 

02563 

32031 

23 

38 

01557 

17363 

35025 

54883 

77365 

03020 

32560 

22 

39 

01807 

17641 

35336 

55235 

77765 

03479 

33090 

21 

40 

02057 

17920 

35649 

55587 

78166 

03938 

33622 

20 

41 

3.02308 

3.18199 

3.35962 

3.55940 

3.78568 

4.04398 

4.34154 

19 

42 

02559 

18479 

36276 

56294 

78970 

04860 

34689 

18 

43 

02810 

18759 

36590 

56649 

79374 

05322 

35224 

17 

44 

03062 

19040 

36905 

57005 

79778 

05786 

35761 

16 

45 

03315 

19322 

37221 

57361 

80183 

06251 

36299 

15 

46 

03568 

19604 

37537 

57718 

80589 

06717 

36839 

14 

47 

03821 

19886 

37854 

58076 

80996 

07184 

37380 

13 

48 

04075 

20169 

38171 

58434 

81404 

07652 

37923 

12 

49 

04329 

20453 

38489 

58794 

81813 

08121 

38466 

11 

50 

04584 

20737 

38808 

59154 

82223 

08591 

39012 

10 

51 

3.04839 

3.21021 

3.39128 

3.59514 

3.82633 

4.09063 

4.39558 

9 

52 

05094 

21306 

39448 

59876 

83045 

09535 

40106 

8 

53 

05350 

21592 

39768 

60238 

83457 

10009 

40656 

7 

54 

05607 

21878 

40089 

60601 

83871 

10484 

41206 

6 

55 

05864 

22165 

40411 

60965 

84285 

10960 

41759 

5 

56 

06121 

22452 

40734 

61330 

84700 

11437 

42312 

4 

57 

06379 

22740 

41057 

61695 

85116 

11915 

42867 

3 

58 

06637 

23028 

41381 

62061 

85533 

12394 

43424 

2 

59 

06896 

23317 

41705 

6-24-28 

85951 

12875 

43982 

1 

60 

07155 

23607 

42030 

62796 

86370 

13357 

44541 

0 

/ 

19° 

18° 

17° 

16° 

15° 

14° 

13° 

/ 

COSECANTS. 

318       XI.— NATURAL  SECANTS  AND  COSKCANTS. 


/ 

SECANTS. 

/ 

77° 

78° 

79° 

80° 

81° 

82° 

83° 

0 

4.44541 

4.80973 

5.24084 

5.75877  6.39245 

7.18530 

8.20551 

60 

1 

4510:2 

81633 

24870 

76829 

40422 

20020 

2-2500 

59 

•2 

45664 

82294 

25658 

77784 

41002 

21517 

24457 

58 

3 

46228 

82956 

26448 

78742 

42787 

23019 

26425 

57 

4 

46793 

83621 

272  41 

79703 

43977 

24529 

28402 

56 

5 

47360 

84288 

28036 

80667 

45171 

26044 

303S8 

55 

6 

4T9i8 

84956 

28883 

81635 

46369 

27566 

3-2384 

54 

48498 

85627 

29634 

88606 

47572 

29095 

34390 

53 

8 

49069 

86299 

30436 

83581 

48779 

30630 

36405 

52 

9 

49642 

86973 

31241 

84558 

49991 

32171 

38431 

51 

10 

50:216 

87649 

3:2049 

85539 

51208 

33719 

40406 

50 

11 

4.50791 

4.88327 

5.32859 

5.86524 

6.52429 

7.35274 

8.42511 

49 

12 

51368 

89007 

33071 

87511 

53655 

36835 

44566 

48 

18 

51917 

89689 

34486 

88502 

54886 

3S403 

46632 

47 

14 

52527 

90373 

35304 

89497 

56121 

39978 

48707 

46 

15 

53109 

91058 

36124 

90495 

57361 

41560 

50793 

45 

16 

53692 

91746 

36947 

91496 

58606 

43148 

52889 

44 

17 

54277 

9:2436 

37772 

92501 

59855 

44743 

5491)6 

43 

18 

54863 

93128 

38600 

93509 

61110 

46346 

57113 

4.2 

19 

55451 

93821 

39430 

94521 

62369 

47955 

59241 

41 

20 

56041 

94517 

402ii3 

95536 

63033 

49571 

61379 

40 

21 

4.56632 

4.95215 

5.41099 

5.9(5555 

6.64902 

7.51194 

8.635-28 

39 

22 

57224 

95914 

41937 

97577 

66176 

52825 

65688 

38 

23 

57819 

96616 

42778 

98603 

67454 

54462 

67859 

37 

24 

58414 

97320 

43622 

90633 

68738 

56107 

70041 

36 

25 

59012 

98025 

444(58 

6.00666 

70027 

57759 

72234 

35 

20 

59611 

98733 

45317 

01703 

71321 

59418 

74438 

34 

27 

60:211 

99443 

46169 

02743 

72620 

61085 

76653 

33 

28 

60813 

5.00155 

47023 

03787 

73924 

62759 

78880 

32 

29 

61417 

00869 

47881 

04884 

75233 

64441 

81118 

31 

30 

62023 

01585 

48740 

05886 

76547 

66130 

83367 

30 

31 

4  62630 

5.02303 

5.49603 

6.06941 

6.77866 

7.67826 

8.85628 

29 

32 

63238 

03024 

50468 

08000 

79191 

69530 

87901 

28 

33 

63849 

03746 

51337 

09062 

805-21 

71242 

90186 

27 

34 

64461 

04471 

52208 

10129 

81  806 

72962 

92482 

26 

35 

65074 

05197 

53081 

11199 

83196 

74689 

94791 

25 

36 

65690 

05926 

53958 

12273 

84542 

76424 

97111 

24 

37 

66307 

06657 

54837 

13350 

85893 

78167 

99114 

23 

38 

66925 

0739C 

55720 

14432 

87250 

79918 

9.01788 

22 

39 

67545 

08125 

56605 

15517 

88612 

81677 

04146 

21 

40 

68167 

08863 

57493 

16607 

89979 

83443 

06515 

20 

41 

4.68791 

5.09602 

5.58383 

6.17700 

6.91352 

7.85218 

9.08897 

19 

42 

69117 

10344 

59277 

18797 

92131 

87001 

11292 

18 

43 

70044 

11088 

60174 

19898 

94115 

88792 

13699 

17 

44 

70673 

11835 

61073 

21004 

95505 

90592 

16120 

16 

45 

71303 

1258? 

61976 

221  13 

96900 

92400 

18553 

15 

46 

71935 

13334 

62881 

23226 

98301 

94216 

20999 

14 

47 

72569 

14087 

63790 

24343 

99708 

96040 

23459 

13 

48 

73205 

14842 

64701 

25464 

7.01120 

97873 

25931 

12 

49 

73843 

15599 

65616 

26590 

02538 

99714 

28417 

11 

50 

74482 

16359 

66533 

27719 

03982 

8.01565 

30917 

10 

51 

4.75123 

5.17121 

5.67454 

6.28853 

7.05392 

8.034-23 

9.33130 

9 

52 

75766 

17886 

68377 

29991 

06828 

05291 

35957 

8 

53 

76411 

18652 

69304 

31133 

08-269 

07167 

38497 

7 

54 

77057 

19421 

70234 

32279 

09717 

09052 

41052 

6 

55 

77705 

20193 

71166 

33429 

11171 

10946 

43620 

5 

56 

78355 

209(50 

72102 

34584 

12630 

12S49 

46203 

4 

57 

79007 

21742 

73041 

35743 

14096 

14760 

48SOO 

3 

53 

79661 

225-21 

73983 

36906 

15568 

16681 

51411 

2 

59 

80316 

23301 

74929 

38073 

17046 

18(512 

54037 

1 

GO 

80973 

24084 

75877 

39245 

18530 

20551 

56677 

0 

( 

12° 

tl° 

10° 

9° 

8° 

4  ° 

o 

/ 

COSECANTS. 

XI.— NATURAL  SECANTS  AND  COSECANTS.        319 


/ 

SECANTS. 

, 

84° 

85° 

80° 

87° 

88° 

8«J° 

0 

9.56677 

11.47371 

14.33559 

19.10732 

28.65371 

57.29809 

60 

1 

59333 

51199 

31)547 

21397 

89440 

58.26976 

59 

2 

020C2 

55052 

455H6 

32)82 

29.13917 

59.27431 

58 

3 

64687 

58932 

51676 

43088 

38812 

60.31411 

57 

4 

67387 

62837 

57817 

54119 

64137 

61.39105 

56 

5 

70103 

66769 

64011 

65275 

89903 

62.50715 

55 

6 

72833 

70728 

70258 

70500 

30.10120 

63.60460 

54 

7 

75579 

74714 

76558 

87976 

42802 

64.80572 

53 

8 

78341 

78727 

82913 

99524 

69960 

66.11304 

52 

9 

81119 

82768 

89323 

20.11208 

97607 

67.40927 

51 

10 

83912 

86837 

95788 

23028 

31.25758 

08.75736 

50 

11 

9.86722 

11.90934 

15.02310 

20.34989 

31.54425 

70.16047 

49 

12 

89547 

95060 

08890 

47093 

83623 

71.62285 

48 

13 

92389 

99214 

15527 

59341 

32.13366 

73.14583 

47 

14 

95248 

12.03397 

22223 

71737 

43671 

74.73586 

46 

15 

98123 

07610 

28979 

84283 

74554 

76.39655 

45 

16 

10.01015 

11852 

35795 

96982 

33.06030 

78.13274 

44 

17 

03923 

16125 

42672 

21.09838 

38118 

79.94908 

43 

18 

00849 

20427 

49611 

22852 

70835 

'  81.85315 

42 

19 

09792 

24761 

56614 

36027 

34.04199 

83.84947 

41 

20 

12752 

29125 

63679 

49368 

38232 

85.94561 

40 

21 

10.15730 

12.33521 

15.70810 

21.02876 

34.72952 

88.14924 

39 

22 

18725 

37948 

78005 

76555 

35.08380 

90.46886 

38 

23 

21739 

42408 

85268 

90409 

44539 

92.91:387 

37 

24 

24770 

46900 

92597 

22.04440 

81452 

95.49471 

36 

25 

27819 

51424 

9H995 

18653 

36.19141 

98.22303 

35 

26 

30887 

55982 

16.07462 

33050 

57633 

101.11185 

34 

27 

33973 

60572 

14999 

47035 

90953 

104.17574 

33 

28 

37077 

05197 

22607 

62413 

37.37127 

107.43114 

32 

29 

40201 

69856 

30287 

77386 

78185 

110.89656 

31 

30 

43343 

74550 

38041 

92559 

38.20155 

114.59301 

30 

31 

10  46505 

12.79278 

16.45869 

23.07935 

38.63008 

118.54440 

29 

32 

49685 

84042 

53772 

23520 

39.06957 

122.77803 

28 

33 

52SS6 

88841 

61751 

39316 

51855 

127.32526 

27 

31 

56106 

93617 

69808 

553-29 

97797 

132.22229 

26 

35 

59346 

9S549 

77944 

71563 

40.44820 

137.51108 

25 

3(i 

62005 

13.03458 

86159 

88022 

92903 

143.24061 

24 

3r 

65885 

08040 

91456 

24.04712 

41.42206 

149  46837 

23 

38 

69186 

13388 

17.02835 

21637 

92772 

156.26228 

22 

3'J 

72507 

18411 

11297 

38802 

42.44525 

103.70325 

21 

40 

75849 

23472 

19843 

56212 

97571 

171.88831 

20 

41 

10.79212 

13.28572 

17.28476 

24.73873 

43.5196! 

180.93496 

19 

42 

82506 

33712 

37196 

91790 

44.07746 

190.98680 

18 

43 

80001 

38891 

40005 

25.09969 

64980 

202.22122 

17 

41 

89428 

44112 

54903 

28414 

45.23720 

214.85995 

16 

4*} 

92877 

49373 

G3S93 

47134 

84026 

229.18385 

15 

46 

96348 

5407'6 

72975 

00132 

46.45963 

245.55402 

14 

47 

99841 

60021 

82152 

85417 

47.09590 

264.44269 

13 

48 

11.03356 

65408 

91424 

26.04994 

74997 

286.47948 

12 

49 

06894 

70838 

18.00794 

24869 

48.42241 

312.52297 

11 

50 

10455 

76312 

10262 

45051 

49.11406 

343.77516 

10 

51 

11.14039 

13.S1H29 

18.19830 

26.65546 

49.82576 

381.97230 

9 

52 

17646 

87391 

29501 

86360 

50.558  40 

429.71873 

8 

S3 

21277 

92999 

39274 

27  07503 

51.31290 

491.10702 

7 

54 

24932 

98651 

49153 

28981 

52.09027 

572.95809 

6 

55 

28610 

14.04350 

59139 

50804 

89156 

6S7.54960 

5 

56 

32313 

10096 

C9233 

72978 

53.71790 

859.43689 

4 

57 

30040 

15889 

79438 

95513 

54.57046 

114').  9157 

3 

58 

:J/.)7'.)2 

21730 

89755 

28.18417 

55.45053 

1718.8735 

2 

59 

43569 

27620 

19.00185 

41700 

56.35946 

3437.7468 

1 

60 

47371 

33559 

10732 

65371 

57.29869 

QO 

0 

/ 

5° 

4° 

3° 

2° 

1° 

0° 

/ 

COSECANTS. 

320 


TABLE  XII. -TANGENTS  AND  COTANGENTS. 


0° 

1°          n          2°                     3°          !  , 

Tang 

Cotang 

Tang 

Cotang  ]    Tang 

Cotang 

Tang   1  Cotang  [ 

0 

.00000 

infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811    60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755    59 

2 

.00058 

1718.87 

.01804 

65.4415 

.03550 

28.1664 

.05299 

18.8711  '58 

3 

.00087 

1145.92 

.01833 

54.5013 

.03579 

27.9372 

.05328 

18.7678    57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656    56 

5 

.00145 

687.549 

.01cfJ91 

52.8821 

.03638 

27.4899 

.05387 

18.5645    55 

0 

.00178 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645    54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

87.0566 

.05445 

18.3655    53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.0547'4 

18.2677    52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708  '51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.47'8 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863    48 

18 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934    47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6118 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

10 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

ir 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

4:2 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7'185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

2:2 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990    38 

28 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.915G    37 

24 

.00698 

143.237 

.02444 

40.9174 

.•04191 

23.8593 

.05941 

16.8319 

30 

25 

.0072? 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

,02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075  !32 

:.'!* 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

82 

.00981 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952  |28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190    27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

85 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

:-5i; 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

21 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

:-',s 

.01105 

90.4633 

.02851 

35.0(595 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

2D 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

43 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.0649(5 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

83.0452 

.04774 

20.9460 

.06525 

15.3254 

1(5 

45 

.01309 

76.3900 

.03055 

32.7?.J3 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.ciass 

74.7292 

.03084 

32.4,13 

.04833 

20.6932 

.06584 

15.1893 

14 

/47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.0(5613 

15.1222 

i:i 

48 

.01396 

71.6151 

.03143 

81.8305 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.08259 

30.6833 

.051)07 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

<52.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.051.53 

19.4051 

.06905 

14.4823 

8 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.08968 

14.3607 

1 

60 

.01746 

57.2900 

.08498 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

/ 

Cotangj    Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

r 

89° 

88° 

87' 

86° 

• 

TABLE  XII. -TANGENTS  AND  COTANGENTS. 


321 


r 

| 

; 

i 

)° 

( 

5° 

•3 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

i 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

0 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.123;38 

8.10536 

58 

3 

.07U80 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.106.28 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

ti 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599  : 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

.93438 

49 

12 

.07:344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

.89734 

47 

If 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

.80622 

42 

1!) 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

.78825 

41 

20 

.07578 

13.1969 

.09:335 

10.7119 

.11099 

9.00983 

.12869 

.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

.73480 

38 

2;] 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

.71715 

37 

21 

.07UU3 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.868C2 

.13047 

.66466 

34 

27 

.07782 

12.8496 

.C9541 

10.4813 

.11305 

8.84551 

.13076 

.64732 

38 

2H 

.07812 

12.8014 

.09570 

10.4491 

'    .11335 

8.82252 

.13106 

.63005 

32 

2'J 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

.59575 

30 

31 

.07899 

12.6591 

.09658 

10.a538 

!    .11423 

8.75425 

.13195 

.57872 

29 

;;•: 

.07929 

12.6124 

.09688 

10.3224 

.114E2 

8.7-31:2 

.13224 

.56176 

28 

:;:; 

.07958 

12.5660 

.09717 

10.2913 

I    .11482 

8.70931 

.13254 

.54487 

27 

3* 

.07987 

12.5199 

.09746 

10.2602 

S   .11511 

8.68701 

.13284 

7.52806 

•26 

85 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

-5 

86 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

87 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

8fi 

.08104 

12.3390 

.09P64 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

:<o 

.08134 

12.2946 

.09893 

10.1080 

;  .ii'  ' 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.0995,9 

10.0483 

.11718 

F.  63402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

40 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

40 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

d~/ 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

•/8 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7&53 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

i   .13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24:345 

.13906 

7.19125 

5 

50 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

,12190 

8.20352 

.13965 

7.16071 

3 

58 

.08(590 

11  5072 

.10452 

9.56791 

,12219 

8.18370 

,13995 

7.14553 

2 

5!) 

.08720 

11.4685 

.10481 

9.54106 

,12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.  12278 

8.14435 

.14054 

7.11537_ 

0 

/ 

Cotang 

Taiig 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

8 

5° 

8 

4' 

S 

3°           1 

8 

2° 

322 


TAUi,E  XIL-'i'ANUENTS  AND  COTANGENTS. 


8° 

9° 

10°           ||           IP           | 

Tang  |  Cotang 

!  Tang 

Cotang 

Tang 

Cotang 

Tang  |  Cotang 

0 

.14054 

7.11537 

.15!S;J8 

6.31375 

.17633 

5.07128 

.19438     5.14455 

60 

1 

.14084 

7.100:38 

.15868 

6.30189 

.17003 

5.00165 

.19408 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17093 

5.05205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.04248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.03295 

.19559 

5.11279 

50 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19019 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.  14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19080 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.00584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.  042!  .7 

47 

14 

.14470 

6.91104 

!   .16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

1C 

.14529 

6.8827'8 

.10316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

1? 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.10370 

6.10664 

.18173 

5.50204 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.40648 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.10525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.10555 

6.04051 

.isar)3 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.  75838 

.16585 

6.02962 

.18384 

5.43906 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

a^; 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

*2 

29 

.14915 

6.70450 

.16704 

5.98016 

.J8504 

5.40429 

.20315 

4.92249 

31  | 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.m564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66403 

.16794 

5.95448 

.18f><)4 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.10824 

5.94390 

.18024 

5.36936 

.20436 

4.89a30 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.  18654 

5.36070 

.20406 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18084 

5.35206 

.20497 

4.87882 

25 

30 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.3177'8 

.20018 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20048 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

40 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19100 

5.2:«m 

.20921 

4.77978 

11 

50 

.15640 

6.43484 

.17333 

5.76937 

.19136 

5.22506 

.20952 

4.77.286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

80 

J5888 

6.31375 

.17633 

5.67128 

.19138 

5.14455 

.21256 

4.70463 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotaii£ 

Tang 

Cotaug 

Tang 

/ 

81°           i            80° 

79° 

78° 

TABLE  XII.— .TANGENTS  AND  COTANGENTS. 


323 


12° 

13° 

14°                ! 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang  |  Cotang 

Tang  |  Cotang 

0 

.21256 

4.70463 

'.23087' 

4.33148 

.24933 

4.01078 

.26795     3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826     3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995  i  4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.  31430 

.25026     3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056     3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240  !  4.30291 

.25087     3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.215£9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.91232 

.27232 

3.67217 

40 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

i8 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25581 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

2-3 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

2:>> 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

20 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.277W 

3.60996 

81 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3  60588    30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

33 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

2s 

38 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14*405 

.25986 

3.84824 

.27858 

3.58966 

20 

86 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

87 

.22383 

4.46764 

.24223 

4.12825 

26079 

3.83449 

.27952 

3.57758 

28 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3  82992 

.27983 

3.57357 

22 

:51) 

.22444 

4.45548 

.2-1285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.21316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

13 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

,26297 

3.80276 

.28172 

3.54968 

10 

45     .22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46;   .22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47  i   .22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48  !   .22719 

4.40152 

.24562 

4.07127 

.26421 

8.78485 

.28297 

3.53393 

12 

49j   .22750 

4.39560 

.21593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.32811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

5;2 

.22842 

4.37793 

.24686 

4.05092 

.26516     3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577     3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608     3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639  !  8.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

,26795 

3.73205 

.28075 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang     Tang 

Cotang 

Tang 

/ 

77° 

i           76°           II           75° 

74° 

324 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


1 

6° 

1 

1 

8°               ! 

1 

9° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.287:38 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

a 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.00857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2  87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

19 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.3-1954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

IS 

.29242 

3.41973  1 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.&3169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

87 

24 

.29432 

3.39771 

.31338 

3.19100 

.332G6 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.W  II  Hi 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

2'5 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

88 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.  9!  1447 

.35346 

2.82914 

32 

29 

.29590 

3.371)55 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

81 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.  33557 

2.98004 

.a5510 

2.81610 

•27 

34 

.29748 

3.36158 

.31658 

3.15877 

'.33589 

2.97717 

.35543 

2.81350 

36 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

•25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.3508? 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

•-•3 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

& 

.7.) 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

;.'! 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

11 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

IS 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.9487'2 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

4(i 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.S5969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

•  2 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

Jl 

50 

.30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36K34 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.343% 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

8.277(57 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90(i96 

.36364 

2.74997 

1 

00 

.30573 

3.27085 

.3:M<)2 

3.07768 

.34433 

2.90421 

.86397 

2.74748 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Cotaug 

Tang 

/ 

7 

3° 

7 

2° 

7 

7 

0° 

TABLE  m— TANGENTS  AND  COTANGENTS. 


325 


20°            !           21° 

22° 

23° 

Tang     Cotang 

Tang 

Cotang 

Tang 

Cotang  '    Tang  |  Cotang 

0     .30397  i  2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1     .83430 

2.74499 

.33420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2     .36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3    .36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4     .3(5329 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5,    .36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6:   .36595 

2.732G3 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7j   .36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8     .36661 

2.72771 

.38654 

2.587'08 

.40674 

2.45860 

.42722 

2.34069 

52 

9     .36694 

2.72526 

.38687 

2.58484 

.40707 

2.45055 

427'57 

2.33881 

51 

10     .36727 

2.72281 

.38721 

2.58261 

.407'41 

2.45451 

!427;91 

2.33693 

50 

11     .36760 

8.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42800 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71082 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70319 

.38921 

2.50928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19j    .370:24 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.G9853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

!   .39122 

2.55008 

.41149 

2.43019 

.43205 

2.31456 

3S 

23 

.37157 

2.69131 

1   .39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

!    .39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2  67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.5:3432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

85 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

:]f> 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2,52357 

.41600 

2.40038 

.43724 

2.28710 

23 

38 

.37054 

2.65576 

.39860 

2.52142 

.41094 

2.39841 

.43758 

2.28528 

22 

:M» 

.37687 

2.65342 

.39694 

2.519:29 

.41728 

2.39645 

.43793 

2.28348  :21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64612 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

41899 

2.38668 

.43966 

2.27447  |16 

45 

.37887 

2.63915 

.39896 

2.50052 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50410 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.20909    13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

S.  49807 

.42070 

2.37097 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098     2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62:332 

.40132 

2.49177 

.42173 

2.37118 

.442-14 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.367-33 

.44314 

2.25663 

0 

55 

.38220 

2.61646 

.40234      2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267     2  48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301      2.48132 

.42345 

2.36158 

.44418 

2.25132 

8 

:,s 

.38320 

2.60963 

.40335      2.47924 

.42379 

2.35967 

.44453 

2.24956 

59 

.38353 

2.60736 

.40369      2.47716 

.42413 

2.35776 

.44488     2.24780 

1 

60 

.38386 

2.60509 

.40403  :  2.47509  | 

.42117 

2.35585 

.44523  |  2.24604 

0 

t  ,  Cotang 

Tang 

Cotang      Tang 

Cotang      Tang 

Cotang     Tang 

] 

1           69° 

68* 

67°           li           66° 

i 

326 


TABLE  xir.— TANGENTS  AND  COTANGENTS. 


/ 

24° 

25°           i            26° 

27°             | 

Tang 

Cotang 

Tang 

Cotang  |   Tang  |  Cotang 

Tang 

Cotang 

0 

.44523 

2.24604 

.46631 

2.14451 

.48773     2.05030 

.50953      1.962I51    60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989      1.96120    59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026      1.95979  :58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838  157 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698  '56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557    55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417    54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277    53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137    52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03875 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.027'80 

.51503 

1.94162    45 

10 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023    44 

1? 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885    43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746    42 

19 

.45187 

2.21304 

,4r305 

2.11392 

.49459 

2.02187 

.51651 

1.93608    41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.207'90 

.47412 

2.10916 

.49568 

2.017'43 

.51761 

1.93195 

38 

98 

.45327 

2.20619 

.47448 

2.10758 

|    .49604 

2.01596 

.51798 

1.93057    37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49610 

2.01449 

.51835 

1.92920    36 

26 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.497'49 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.922.35 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

:« 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

i    .50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

.2.08560 

[   .50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39     .45SS!) 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741    20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607  'l9 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472  1  18 

43  1   .46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337    17 

44     .46065 

2.17083 

.48198 

2.07476 

.50368     1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404      1.98396 

.52613 

1.90069 

15 

46 

.46136     2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935    14 

47 

.46171     2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.8S801  !l3 

48 

.4620(5 

2.16420 

.48342 

2.06860 

.50514 

1.97'966 

.52724 

1.89667  j  18 

4!) 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533    11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.8926b 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56     .46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.58023 

1.88602 

4 

57     .46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58     .46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

59     .46595 

2.14614 

.48737 

2.0.->1S2 

.50916 

l!96402 

.53134 

L8820B 

1 

CO     .46631 

2.14451 

.48773  i  2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

f 

Cotantf 

Tang 

Cotang     Tang 

Coum- 

Tang 

Cot  aii  g 

Tang 

/ 

65°           1 

64° 

63° 

62°          1 

TABLE  X1I.-TANOENTS  AND  COTANGENTS. 


32? 


2 

5° 

2 

9° 

3 

jo 

3 

1° 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

-o 

.53171 

.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

.87809  i 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

.87283 

.55659 

1.79665 

i   .57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

.87021 

.55736 

1.79419 

1   .58046 

1.72278 

.60403 

1.65554 

52 

9 

53507 

.86891 

.55774 

1.79296 

!   .58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

.86780 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

.86369  ! 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

.86239 

.55964 

1.78685 

.58279 

.71588 

.60642 

1.64903 

46 

15 

.53732 

.86109 

.56003 

1.78563 

.58318 

.7147'3 

.60681 

1.64795 

45 

16 

.53769 

.85979 

.56041 

1.78441 

.58357 

.71358 

.60721 

1.64687 

44 

17 

.53807 

.85850 

.56079 

1.78319 

.58396 

.71244 

.60761 

1.64579 

43 

18 

.53844 

.85720 

.56117 

1.78198 

.58435 

.71129 

.60801 

1.64471 

42 

19 

.53882 

.85591 

.56156 

1.78077 

.58474 

.71015 

.60841 

1.64363 

41 

20 

.53920 

.85462 

.56194 

1.77955 

.58513 

.70901 

.60881 

1.64256 

40 

21 

.53957 

.85333 

.56232 

1.77834 

.58552 

.70787 

.60921 

1.64148 

39 

22 

.53995 

.85204 

.56270 

1  .77713 

.58591 

.70673 

.60960 

1.64041 

m 

23 

.54032 

.85075 

.56309 

1.77592 

.58631 

.70560 

.61000 

1.63934 

37 

24 

.54070 

.84946 

.56:347 

1.77471 

.58670 

.70446 

.61040 

1.63826 

36 

25 

.54107 

.84818 

.56385 

1.77351 

.58709 

.70332 

.61080 

1.63719 

35 

20 

.54145 

.84689 

.56424 

1.77230 

.58748 

.70219 

.61120 

1.63612 

34 

27 

.54183 

.84561 

.56462 

1.77110 

.58787 

.70106 

.61160 

1.63505 

33 

28 

.54220 

.84433 

.56501 

1.76990 

.58826 

.69992 

.61200 

1.63398 

32 

29 

.54258 

.84305 

.56539 

1.76869 

!   .58865 

.69879 

.61240 

1.63292 

31 

80 

.54296 

.84177 

.56577 

1.76749 

|    .58905 

.69766 

.61280 

1.63185 

30 

31 

.54333 

.84049 

.56616 

1.76629 

.58944 

.69653 

.61320 

1.63079 

29 

32 

.54371 

.83922 

.56654 

1.76510 

i   .58983 

.69541 

.61360 

1.62972 

28 

33 

.54409 

.&3794 

.56693 

1.76390 

.59022 

.69428 

.61400 

1.62866 

27 

34 

.54446 

.83667 

.56731 

1.76271 

.59061 

.69316 

.61440 

1.62760 

26 

35 

.54484 

.83540 

.56769 

1.76151 

.59101 

.69203 

.61480 

1.62654 

25 

36 

.54522 

.83413 

.56808 

1.76032 

.59140 

.69091 

.61520 

1.62548 

24 

37 

.54560 

.83286 

.56846 

1.75913 

j   .59179 

.68979 

.61561 

1.62442 

23 

38 

.54597 

.&3159 

.56885 

1.75794 

i    .59218 

.68866 

.61601 

1.62336 

22 

39 

.54635 

.83033 

.56923 

1.75675 

.59258 

.68754 

.61641 

1.62230 

21 

40 

.54673 

.82906 

.56962 

1.7'5556 

.59297 

.68643 

.61681 

1.62125 

20 

41 

.54711 

.82780 

.57000 

1.75437 

.59336 

.68531 

.61721 

1.62019 

19 

42 

.54748 

.82654 

.57039 

1.75319 

i   .59376 

.68419 

.61761 

1.61914 

18 

43 

.54786 

.82528 

.57078 

1.75200 

.59415 

.68308 

.61801 

1  61808 

17 

44 

.54824 

.82402 

.57116 

1.75082 

.59454 

.68196 

.61842 

1.61703 

16 

45 

.54862 

.82276 

.57155 

1.74964 

.59494 

.68085 

.61882 

1.61598 

15 

4(5 

.54900 

.82150 

.57193 

1.74846 

.59533 

.67974 

.61922 

1.61493 

14 

47 

.54938 

.82025 

.57232 

1.747'28 

.59573 

.67863 

61962 

1.61388 

18 

48 

.54975 

.81899 

.57271 

1.74610 

.59612 

.67752 

.62003 

1.61283 

12 

49 

.55013 

.81774 

.57309 

1.74492 

.59651 

.67641 

.62043 

1.61179 

11 

50 

.55051 

.81649 

.57348 

1.74375 

.59691 

.67530 

.62083 

1.61074 

10 

51 

.55089 

.81524 

.57386 

1.74257 

.59730 

.67419 

.62124 

1.60970 

9 

52 

.55127 

.81399 

.57425 

1.74140 

.59770 

.67309 

.62164 

1.60865 

8 

53 

.55165 

.81274 

.57464 

1.74022 

i   .59809 

.67198 

.62204 

1.60761 

y 

54 

.55203 

.81150 

.57503 

1.73905 

.59849 

.67088 

.62245 

1.60657 

6 

55 

.55241 

.81025 

.57541 

1.73788 

.59888 

.6697'8 

.62285 

1.60553 

5 

5fi 

.55279 

.80901 

.57580 

1.73671 

.59928 

.66867 

.62325 

1.60449 

4 

57 

.55317 

.80777 

.57619 

1.73555 

.59967 

.66757 

.62366 

1.60345 

3 

58 

.55355 

.80653 

.57657 

1.73438 

.60007 

.66647 

.62406 

1.60241 

2 

59 

.5;>3<)3 

.80529 

.57696 

1.73321 

.60046 

.66538 

.62446 

1.60137 

1 

60 

.55431 

.H0405 

j   .57735 

1.73205 

.60086 

.66428 

.024S7 

1.60033 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

"f 

€ 

i° 

6 

0° 

5 

9°           ! 

5 

8° 

TABLE  XII.-TANGENTS  AND  COTANGENTS. 


32°                        33° 

34° 

35° 

Tang  |  Cotang   | 

Tang 

Cotang 

Tang     Cotang 

Tang 

Cotang 

0!  .62487 

1.000:3:3 

.04941 

1.53986 

.07451 

1.48256 

.70021 

1.42815 

50 

1     .62527 

1.  59SK30 

.64982 

1.53888 

.67493 

1.48163 

.70004 

1.42726 

59 

2     .62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.05005 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885  1 

.70194 

1.42462 

56 

5 

.62689 

1.59537 

.65148 

1.53497 

.67603 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47099 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.05231 

1.53302 

.67748 

1.47607  ; 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.5:3205 

.67790 

1.47514 

.70308 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422  : 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.6787'5 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70029 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70073 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52332  ! 

.68173 

1.46686 

.70700 

1.41322 

43 

18 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.08258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46471 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946  ! 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.0*386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65938 

1.51658 

.68471 

1.40046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40027 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

•14 

27 

.63584 

1.572"! 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.456S2 

.71242 

1.40367 

32 

2!) 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.50067 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56406 

.66398 

1.50007 

.68942 

1.45049 

71549 

1.39764 

25 

36 

.63953 

1.56300 

.66440 

1.50512 

.68985 

1.44958 

71593 

1.39079 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

71637 

1.39593 

23 

38 

.04035 

1.56165 

.66524 

1.50322 

.09071 

1.44778 

71681 

1.39507 

22 

39 

.64076 

1.56005 

.66500 

1.50228 

.69114 

1.44688 

71725 

1.39121 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

71769 

1.39336 

20 

41 

.6-1158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55706 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55606 

.667'34 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64231 

1.55567 

.6677'6 

1.49755 

.09329 

1.44239 

.71946 

1.38994 

10 

45 

.64322 

1.55467 

.66818 

1.49661 

.09372 

1.44149 

.71990 

1.38909 

15 

48 

.64303 

1.55308 

.66800 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64104 

1.55269 

.66902 

1.49472 

.69459 

1.43970    i    .72078 

1.38738 

13 

48 

.64416 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72123 

1.38053 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38508 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

t 

54 

.64010 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

;    .67155 

1.48909 

.69718 

1.  1:5  r;r. 

!73344 

1.38229 

7 

54 

.64693 

1.54576 

•   .67197 

1.48816 

.69761 

1.4*347 

.72388 

1.38145 

6 

58 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.88060 

5 

6< 

.04775 

1.54379 

.67282 

1.48629 

.69847 

1.43109 

.72477 

1.37970 

i 

5' 

.64817 

1.54281 

;    .67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

{ 

.-> 

.64858 

1.54183 

.07300 

1.48442 

.09934 

1.42992 

.72505 

1.37807 

i 

•  r 

)    .64399 

1.54085 

.67409 

1.4&349       .69977 

1.42903 

.72010 

1.37722 

• 

& 

)     .64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37038 

0 

/ 

Cotang     Tang 

Cotang 

Tang      Cotang  I    Tang      C'otun^ 

Tang 

i 

57°                       56°                      55°            !           54° 

TABLE  m -TANGENTS  AND  COTANGENTS. 


329 


36°           j            37°           i            38°                        39° 

/ 

Tang  |  Cotang 

Tang  |  Cotang  j 

Tang 

Cotang      Tang     Cotang 

0 

.72654  l  1.37638  i 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1  .-27764 

.81123 

1.23270 

57 

4 

.72S32 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

IS 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13     .735234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

161   .73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17    .73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18     .73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19!    .73502 

1.36051 

.76226 

1.31190  ! 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

:js 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

•js 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

'i.  30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166  128 

m 

.74131 

1.34896 

.76871 

1.30087 

.79686 

1.25492 

.82580 

1.21094    27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

89 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77885 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024  Il2 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.3*349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.  75037 

1.33268 

.77801 

1.28533 

.80(542 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.38379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

,83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761  i  1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811  '  1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.33490 

.83910  i  1.19175 

0 

f   Cotang 

Tang 

Cotang 

Tang 

Cotang  j    Tang 

Cotang     Tang 

I           53° 

52° 

1           51° 

1           50° 

330 


TABLE  Xlt.-TANGENTS  AND  COTANGENTS. 


40° 

41°           •            42°           ||           43° 

Tang   !  Cotang 

Tang     Cotang 

Tang 

Cotang 

Tang   •  Cotang 

0 

.83910 

1.19175 

.86929      1.15037 

.90040 

1.11061 

.93252      1.07237 

60 

1 

.83960 

1.19105 

.86980      1.14969  ; 

.90093 

1  .  10996 

.93306        .07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14834 

.90199 

1.10807 

.93415 

.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

.06987 

50 

E 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1  .  10737 

.93524 

.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14032 

1  .  10072 

.93578  i     .06802 

54 

7 

.84258 

1.18684 

.87287 

1.14505 

.90410 

1.10607 

.93633        .00800 

53 

8 

,84307 

1.18614 

.87338     1.14498 

.90463 

1.10543 

.93088        .06738 

52 

.84357 

1.18544 

.87389     1.14430 

.90516 

1.10478 

.93742        .06670 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852        .06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906        .06489    48 

13 

.84556 

1.18264 

i   .87595 

1.14162 

.90727 

1.10220 

.93961        .06427    47 

14 

.84606 

1.18194 

1   .87646 

1.14095 

.90781 

1.10150 

.94010        .06365  J46 

15 

.84656 

1  .  18125 

1   .87698 

1.14028 

.90834 

1.10091 

.94071        .06303  !45 

10 

.84706 

1.18055 

.87749  >  1.13961 

.90887 

1.10027 

.94125        .06241  j  44 

17 

.84756 

1.17986 

.87801 

1.1-3894 

.90940 

1.09903 

.94180        .06179    43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

1.09899 

.94235  !     .06117    42 

19 

.84856 

1.17846 

.87'904 

1.13761 

.91046 

1.09834 

.94290 

.06056 

41 

20 

.84906 

1  .  17777 

.87955 

1.13094 

.91099 

1.09770 

.94345 

.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09042 

.94455 

.05870    38 

23 

.85057 

1.17509 

.88110     1.13494 

.91259 

.09578 

.94510 

.05809    37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

.09514 

.94565 

.05747 

36 

25 

.85157 

1.17430 

.88214 

1  .  13361 

.91366 

.09450 

.94620 

.05685 

35 

20 

.85207 

1.17361 

.88265 

1.13295 

.91419 

.09380 

.94676        .05624    34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

.09322 

.94731  I     .05562  ;33 

28 

.85308 

1.17223 

.88369     1.13102  ! 

.91526 

.09258 

.94786 

.05501   :32 

29 

.85358 

1.17154 

.88421 

1.13090  ; 

.91580 

.09195 

.94841 

.05439    31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

.09131 

.94896 

.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963  i 

.91687 

.09067 

.94952  '     .05317    29 

32, 

.85509 

1.16947 

.88576 

1.12897 

.91740 

.09003 

.95007        .05255  i28 

83 

.85559 

1.168T8 

.88628 

1.12831 

.91794 

.08940 

.95062        .05194    27 

34 

.85609 

1.16809 

.88680 

1.12765 

.91847 

.08876 

.95118 

.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699  i 

.91901 

.08813 

.35173 

.05072 

25 

30 

.85710 

1.16072 

.88784 

1.12633 

91955 

.08749 

.95229 

.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

.08686 

.95284 

.04949 

23 

38     .85811 

1  .  16535 

.88888 

1.12501 

.92062 

.08622 

.95340 

.04888    22 

39     .85862 

1.16466 

.88940 

1.12435 

.92116 

.08559       .95395 

.04827  121 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

.08490  j     .95451 

.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

.08432  1     .95506 

.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

.08309    '    .95502 

.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172       .92331 

.08306       .95618 

.04583    17 

44 

.86115 

1.16124 

.89201 

1.12106  '     .92385 

.08243    !   .95673 

.04522  116 

45 

.86166 

1.16056 

.89253 

1.12041  ,     .92439 

.08179 

i   .95729 

.04461  115 

40 

.86216 

1.15987 

.89306 

1.11975       .92493 

.08116 

.95785 

.04401   J14 

47 

.86267 

1.15919 

.89358 

1.11909       .92547 

.08053 

.95841 

.04340    13 

48 

.86318 

1.15851 

.89410 

1.11844  ';    .92601 

•07990 

.95897 

.04279    19 

491    .86368 

1.15783 

.89403 

1  11778  I     .92055 

.07927 

.95952 

.04218    11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

.07864 

.96008 

.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

.07801 

.96064 

.04097 

9 

52     .86521 

1  .  15579 

.89620 

1.11582 

.92817 

.077'38 

.96120 

.04036 

8 

53     .86572 

1.15511 

.89672 

1.11517 

.92872 

.07676 

.96176 

.03976 

7 

54 

.86623 

1  .  15443 

.89725 

1.11452 

.92926 

.07613 

.96232 

.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

.07550 

.96288 

.03855 

5 

56 

.867'25 

1.15308 

.89aso 

1.11321 

.93034 

.07-487 

.96344 

.03794 

4 

57 

.86776 

1  .  15240 

.89883 

1.11256 

.93088 

.07425  • 

.96400 

.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

.07362  i 

.96457 

.08674 

2 

59 

.8687'8 

1.15104 

.899S8 

1.11126 

.93197 

1.07299 

.96513 

.03613 

1 

00 

.86929 

I  !  15087 

.90040      1.11061 

.93252 

1.07237 

.96569 

113553 

0 

f 

Cotang     Tang 

n>t  iing  |    Tang 

Cotang 

Tang      Cotang     Tang 

f 

49°            1           48°            i           47°           li           46° 

TABLE  XII.-TANGENTS  AND  COTANGENTS. 


4 

g 

4 

[4° 

4 

4° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03563 

60 

20 

.97700 

1.02355 

40 

140 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.98901 

.01112 

19 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

38 

42 

.98958 

.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

.00994 

17 

4 

.96794 

.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

.00935 

16 

5 

.96850 

.03252 

55 

25 

.97984 

.02057 

35 

45 

.99131 

.00876 

15 

6 

.96907 

.03192 

54 

26 

.98041 

.01998 

34 

46 

.99189 

.1)0818 

14 

7 

.96963 

.03132 

58 

27 

.98098 

.01939 

33 

47 

.99247 

.00759 

13 

8 

.97020 

.03072 

53 

28 

.98155 

.01879 

32 

48 

.99304 

.00701 

12 

9 

.97076 

.03012 

51 

29 

.98213 

.01820 

31 

49 

.99362 

.00642 

11 

10 

.97133 

.02952 

50 

30 

.98270 

.01761 

30 

50 

.99420 

.00583 

10 

11 

.97189 

.02892 

49 

81 

.98327 

.01702 

29 

51 

.99478 

.00525 

9 

12 

.97246 

.02832 

48 

32 

.98384 

.01642 

28 

152 

.99536 

.00467 

8 

13 

.97302 

.02772 

47 

83 

.98441 

.01583 

27 

53 

.99594 

'  .00408 

7 

14 

.97359 

.02713 

46 

34 

.98499 

.01524 

26 

;54 

.99652 

.00350 

6 

15 

.97416 

.02653 

45 

35 

.98556 

.01465 

25 

!  55 

.99710 

.00291 

5 

16 

.97472 

.02593 

44 

36 

.98613 

.01406 

24 

!56 

.99768 

.00233 

4 

1? 

.97529 

.02533 

43 

37 

.98671 

.01347 

23 

157 

.99826 

.00175 

3 

18 

.97586 

.02474 

42 

38 

.98728 

.01288 

22 

158- 

.99884 

.00116 

a 

10 

.97643 

.02414 

41 

39 

.98786 

.01229 

31 

'59 

.99942 

.00058 

i 

20 

.97700 

.  02:155 

40 

40 

.98843 

.01170 

20 

60 

1.00000 

1.00000 

o 

/ 

Co  tang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

4 

5° 

4 

5° 

4 

5° 

332 


TABLE  xm.-VERsiNEs  AND  EXSECAKTS. 


; 

0-     j|     1- 

2° 

30 

i 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00000 

.00000 

.00015 

.00015 

.00061 

.00061  1 

.00137 

.00137 

0 

1 

.00000 

.00000 

.00016 

.00016 

.00062 

.00062  ! 

.00139 

00139 

1 

2 

.00000 

.00000 

.00016 

.00016 

.00063 

.00063  ! 

.00140 

.00140 

2 

3 

.00000 

.00000 

.00017 

.00017 

.00064 

.00064 

.00142 

.00142 

3 

4 

.00000 

.00000 

.00017 

.00017 

.00065 

.00065 

.00143 

.00143 

4 

5 

.00000 

.00000 

.00018 

.00018 

.00066 

.00066 

.00145 

.00145 

5 

6 

.00000 

.00000 

.00018 

.00018 

.00067 

.00067 

.00146 

.00147 

6 

y 

.00000 

.00000 

.00019 

.00019 

.00068 

.00068 

.00148 

.00148 

7 

8 

.00000 

.00000 

I  .00020 

.00020 

.00069 

.00069  • 

.00150 

.00150 

8 

9 

.00000 

.00000 

I  .00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

10 

.00000 

.00000 

i  .00021 

.00021 

.00071 

.00072 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

.00154 

.00155 

11 

12 

.00001 

.00001 

I  .00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075 

.00075 

.00158 

.00158 

13 

14 

.00001 

00001   .00023 

.00023 

.00076 

.00076 

.00159 

.00159 

14 

15 

.00001 

.00001   .00024 

.00024 

.00077 

.00077 

.00161 

.00161 

15 

16   .00001  !  .00001 

.00024 

.00024 

.00078 

.00078 

.00162 

.00163 

16 

17 

.00001 

.00001 

.00025 

.00025 

.00079 

.00079 

.00164 

.00164 

17 

18 

.00001 

.00001 

.00026 

.00026 

.00081 

.00081 

.00166 

.00166 

18 

19 

.00002 

.00002 

i  .00026 

.00026 

.00082 

.00082 

.00168 

.00168 

19 

20 

.00002 

.00002 

!  .00027 

.00027 

.00083 

.00083 

.00169 

.00169 

20 

21 

.00002 

.00002 

.00028 

.00028 

.00084 

.00084 

.00171 

.00171 

21 

22 

.00002 

.00002 

.00028 

.00028 

.00085 

.00085 

.00173 

.00173 

22 

23 

.00002 

.00002 

.00029 

.00029 

.00087 

.00087 

.00174 

.00175 

23 

24 

.00002 

.00002 

.00030 

.00030 

.00088 

.00088 

.00176 

.00176 

24 

25 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089 

.00178 

.00178 

25 

26 

.00003 

.00003 

.00031 

.00031 

.00090 

.00090 

.00179 

.00180 

26 

27 

.00003 

.00003 

.00032 

.00032 

.00091 

.00091  ! 

.00181 

.00182 

27 

28 

.00003 

.00003 

.00033 

.00033 

.00093 

.00093 

.00183 

.00183 

28 

29 

.00004 

.00004 

.00034 

.00034 

.00094 

.00094 

.00185 

.00185 

29 

30 

.00004 

.00004 

.00034 

.00034 

.00095 

.00095 

.00187 

.00187 

30 

31 

.00004 

.00004 

.00035 

.00035 

.00096 

.00097 

.00188 

.00189 

31 

32 

.00004 

.00004 

.00036 

.00036 

.00098 

.00098 

.00190 

.00190 

32 

33 

.00005 

.00005 

.00037 

.00037 

.00099 

.00099 

.00192 

.00192 

33 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00194 

.00194 

34 

35 

.00005 

.00005 

.00038 

.00038 

.00102 

.00102 

.00196 

.00196 

35 

36 

.00005 

.00005 

.00039 

.00039 

.00103 

.00103 

.00197 

.00198 

36 

37 

.00006 

.00006 

.00040 

.00040 

.00104 

.00104  i 

.00199 

.00200 

37 

38 

.00006 

.00006 

.00041 

.00041 

.00106 

.00106 

.00201 

.00201 

38 

39 

.00006 

.00006 

.00041 

.00041 

.00107 

.00107 

.00203 

.00203 

39 

40 

.00007 

.00007 

.00042 

.00042 

.00108 

.00108 

.00205 

.00205 

40 

41 

.00007 

.00007 

.00043 

.00043 

.00110 

.00110 

.00207 

.00207 

41 

42 

.00007 

.00007 

.00044 

.00044 

.00111 

.00111 

.00208 

.00209 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00046 

.00046 

.00114 

.00114 

.00212 

.00213 

44 

45 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115  i 

.00214 

.00215 

45 

46 

.00009 

.00009 

.00048 

.000-18 

.00117 

.00117 

.00216 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00118 

.00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00049 

.00049 

.00119 

.00120 

.00220 

.00220 

48 

49 

.00010 

.00010 

.00050 

.00050 

.00121 

.00121 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.00053 

.00058 

.00125 

.00125 

.00228 

.00228 

52 

53 

.00012 

.00012 

.00054 

.00054 

.00127 

.00127 

.00*30 

.00230 

53 

54 

.00012 

.00012 

.00055 

.00055 

.00128 

.00128 

.00238 

.00232 

54 

55 

.00013 

.00013 

.00056 

.00056 

.00130 

.00130 

.00234 

.00234 

55 

56 

.00013 

.00013 

.00057 

.00057 

.00131 

.00131 

.00236 

.00236 

56 

57 

.00014 

.00014 

I  .00058 

.00058 

.00133 

.00133 

.00238 

.002:38 

57 

58 

.  It!  11)11 

.oonit 

1  .00059 

.00059 

.00134 

.00134 

.00240 

.00240 

58 

59 

.00015 

.00015 

j  .00060 

.0(X)t;o 

.00136 

.00136 

.00343 

.00242 

59 

60 

.00015 

.00015 

1  .00061 

.00061 

.00137 

.00137 

.00244 

.00244 

60 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


333 


4° 

5C 

6° 

7° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00244 

.00244  ! 

.00381 

.00382  : 

.00548 

.00551 

.00745 

.00751   0 

1 

.00246 

.00246  ' 

.00883 

.00385 

.00551 

.00554 

.00749 

.00755 

j 

2 

.00248 

.00248 

.00886 

.00387 

.00554 

.00557 

.00752 

.00758 

2 

3 

.002-50 

.002.50 

.00388 

.00390 

.00557 

.00560 

.00756   .007'62  i  3 

4 

.00252 

.00252  : 

.00391 

.00392 

.00560 

.0056:-! 

.00760  ''  .00765  !  4 

5 

.00254 

.00254 

.00393 

.00395 

.00563 

.00566 

.00763 

.00769 

5 

6 

.00256 

.00357 

.00396 

.00397 

.00666 

.00569 

.00767 

.00773 

6 

7 

.00258 

.00259 

.00398 

.00400 

.00569 

.00573  , 

.00770 

.00776 

*  7 

8 

.00260 

.00261 

.00401 

.00408  ; 

.00572 

.00576 

.00774 

.00780 

8 

9 

.00262 

.00263 

.0040-1   .00405 

.00576 

.00579  1 

.00778 

.00784 

9 

10 

.00264 

.00265  1 

.00406 

.00408 

.00579 

.00582  I 

.00781   .00787 

10 

11 

.00266 

.00267 

.00409   .00411 

00582 

.oosa5  ; 

.00785 

.00791 

11 

12 

.00269 

.00269  ! 

.00-412 

.00413 

.00585 

.00588 

.00789 

.00795 

12 

13 

.00271 

.00271 

.00414 

.00416  ; 

.00588 

.00592 

.00792 

.00799 

13 

14 

.00273 

.00274 

.00417 

.00419 

.00591 

.00595  1 

00796 

.00802 

14 

15 

.00275 

.00276 

.00420 

.00421 

00594 

..00598  i 

.00800 

.00806 

15 

16 

.00277 

.00278 

.00422   .00424 

.00598 

.00601  I 

.00803 

.00810 

16 

17 

.00279 

.00280 

.00425  i  .00427  : 

.00601 

.00604  ! 

.00807 

.00813 

17 

18 

.00281 

.00282 

.00128   .00429 

00604 

.00608  ; 

.00811 

.00817 

18 

19 

.00284 

.00284 

.00430 

.00432 

.00607 

.00611  i 

.00814 

.00821 

19 

20 

.00286 

.00287  ! 

.00433 

.00435  ; 

.00610 

.00614  ' 

.00818 

.00825 

20 

21 

.00288 

.00289  i 

.00436 

.00438  i 

.006!4 

.00617 

.00822 

.00828 

21 

22 

.00290 

.00291 

.00438 

.00440 

.00617  •  .00621 

.00825 

.00832 

22 

23 

.00293 

.00293 

.00441 

.00443 

.00620   .00624  : 

.00829 

.OOH36 

23 

24 

.00295 

.00296  ! 

.00444 

.00446 

.00623 

.00627  ' 

.00833 

.00840- 

24 

25  .00297 

.00298  ! 

.00447 

.00449 

.00626 

.00630 

.00837  i  .00844 

25 

26 

.00299 

.00300  i 

.00-149 

.00451 

.00630 

.00634 

.00840   .00848 

26 

27 

.00301 

.00302  i 

.00452   .00454  : 

.00633 

.00637 

.00844   .00851 

27 

28  !  .00304 

.00305 

.00155 

.00457 

.00636 

.00640 

.00848   .00855 

28 

29   .00306 

.00307 

.00458 

.00460 

.00640 

.00644 

.00852 

.00859 

29 

30 

.00308 

.00309  ; 

.00460 

.00463 

.00643 

.00647 

.00856 

.00863 

30 

31 

.00311 

.00312 

.00463 

.00465  . 

.00646 

.00650 

.00859   .00867 

31 

32  .00313 

.00314 

.00466 

.00468 

.00649 

.00654 

.OQ863  !  .00871 

32 

33  .00815 

.00316 

.00469 

.00471 

.00(553   .00657 

.00867  ,  .00875 

33 

34 

.00317 

.00318  i 

.00472 

.00474  i 

.00656 

.00660 

.00871   .00878 

34 

35 

.00320 

.0032!  i 

.00474 

.00477 

.00659 

.00664 

.00875   .00882 

35 

36   .00323   .00323 

.00477 

.00480 

.00663   .00667 

.00878  :  .00886 

36 

37  i  .00324   .00326 

.00480 

.00482 

.00666  i  .00671 

.00882  !  .00890 

37 

38   .00327 

.00328 

.00483 

.00485  , 

.00669   .00674  ; 

.00886   .00894 

38 

39 

.00329 

.00330  : 

.00486 

.00488  ; 

.00673   .00677 

.00890   .00898 

39 

40 

.00332 

.00333  : 

.00489 

.00491  i 

.00676 

.00681  : 

.00894   .00902 

40 

41 

.00334 

.00335  : 

.00492 

.004911  ! 

.00680 

.00684 

.00898   .00906 

41 

42 

.00336  I  .00337 

.00494 

.0041)7  : 

.00683   .0(3688 

.00902   .  00910 

42 

43 

.00339   .00340  i 

.00497 

.00500  i 

.00686   .00691 

.mm  .oo9i4 

43 

44 

.00341 

.00312 

.00500 

.00503 

00690   .00695 

.00909  i  .00918 

44 

45 

.00343 

.00345 

.00503 

.00506 

.00693 

.00698    00913   .00922 

45 

46 

.00346 

.00347 

.00506 

.00509 

.00697 

.00701 

00917   .00926 

46 

47 

.00348 

.00350 

.00509 

.00512  i 

.00700 

.00705 

00921  I  .00930 

47 

48 

.00351 

.00352  ' 

.00512 

.00515 

.00703   .00708 

.00925   .00934 

48 

49 

.00353 

.00354  , 

.00515 

.00518 

.00707  !  .00712 

00929 

.00938 

49 

50 

.00356 

.00357  ; 

.00518 

.00521 

.00710 

.00715 

.00933 

.00942 

50 

51 

.00358 

.ooa^g 

.00521 

.00524 

.00714 

.00719 

.00937 

.00946 

51 

52 

.00361 

.003*52 

.00524 

.00527 

.00717   .00722 

.00941 

.00950 

52 

53 

.00363 

.00364 

.00527 

.00530 

.00721  i  .00726 

.00915 

.00954 

53 

54 

.00365 

.00367 

.00530 

.00533 

.00724   .00730 

.00949 

.00958 

54 

55 

.00368 

.00369 

.00533 

.00536 

.00728 

.00735 

.00953 

.00962 

55 

56 

.00370 

.00372 

.00536 

.00539 

.00731 

.00737 

.00957 

.00966 

56 

57 

.00373 

.00374 

.00539 

.00542 

.00785 

.00740 

.00961 

.00970 

57 

58 

.00375 

.00377 

.00542 

.00545 

.00738  ;  .00744 

.00965 

.00975 

58 

59 

.00378 

.00379 

.00545   .00548 

.00742   .00747 

.00969 

.00979 

59 

60 

.00381 

.00382 

.00548   .00551 

.00745   .00751 

.00973 

,00983  60 

334 


TABLE  XIII.— VERSINES   AND   EXSECANTS. 


/ 

8° 

9° 

10° 

11° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00973 

.00983 

.01231 

.01247 

.01519 

.01543  ! 

.01837 

.01872 

0 

1 

.00977 

.00987 

.01236 

.01251 

.01524 

.01548  i 

.01843 

.01877 

1 

2 

.00981 

.00991 

.01240 

.01256 

.01529 

.01553 

.01848 

.oisa3 

2 

3 

.00985 

.00995 

.01245 

.01261 

.01534 

.01558 

.01854 

.01889 

3 

4 

.00989 

.00999 

.01249 

.01265 

.01540 

.01564 

.01860 

.01895 

4 

5 

.00994 

.01004  ! 

.01254 

.01270 

.01545 

.01569 

.01865 

.01901 

5 

6 

.00998 

.01008 

.01259 

.01275 

.01550 

.01574 

.01871 

.01906 

6 

7- 

.01002 

.01012  1 

.01263 

.01279 

.01555 

.01579 

.01876 

.01912 

7 

8 

.01006 

.01016 

.01268 

.01284 

.01560 

.01585 

.01882 

.01918 

8 

9 

.01010 

.01020 

.01272 

.01289 

.01565 

.01590 

.01888 

.01924 

9 

10 

.01014 

.01024 

.01277 

.01294 

.01570 

.01595 

.01893 

.01930 

10 

11 

.01018 

.01029  ! 

.01282 

.01298 

.01575 

.01601 

.01899 

.01936 

11 

13 

.01022 

.01033 

.01286 

.01303 

.01580 

.01606 

.01904 

.01941 

12 

13 

.01027 

.01037 

.01291 

.01308 

.01586 

.01611 

.01910 

.01947 

13 

14 

.01031 

.01041 

.01296 

.01313 

.01591 

.01616 

.01916 

.01953   14 

15 

.01035 

.01046  I 

.01300 

.01318 

.01596 

.01622 

.01921 

.01959  15 

16 

.01039 

.01050  ! 

.01305 

.01322 

.01601 

.01627 

.01927 

.01965 

16 

17 

.01043 

.01051 

.01310 

.01327 

.01606 

.01633 

.01933 

.01971 

17 

18 

.01047 

.01059 

.01314 

.01332 

.01612 

.01638 

.01939 

.01977 

18 

19 

.01052 

.01063  ! 

.01319 

.01337 

.01617 

.01643 

.01944 

.01983 

19 

20 

.01056 

.01067 

.01324 

.01342 

.01622 

.01649 

.01950 

.01989 

20 

21 

.01080 

.01071 

.01329 

.01346 

.01627 

.01654 

.01956 

.01995 

21 

22 

.01061 

.01076 

.01333 

.01351 

.01632 

.01659 

.01961 

.02001 

22 

23 

.01069 

.01080 

.01338 

.01356 

.01638 

.01665 

.01967 

.02007 

23 

24 

.01073 

.01084  : 

.01343 

.01361 

.01643 

.01670 

.01973 

.02013 

24 

23 

.01077 

.01089 

.01348 

.01366 

.01648 

.01676 

.01979 

.02019 

25 

23 

.01081 

.01093 

.01352 

.01371 

.01653 

.01681 

.01984 

.02025 

26 

•>•• 

.01088 

.01097 

.01357 

.01376 

.01659 

.01687 

.01990 

.02031 

27 

28 

.01090 

.01102 

.01362 

.01381 

.01664 

.01692 

.01996 

.02037 

28 

29 

.01094 

.01106 

.01367 

.01386 

.01669 

.01698 

.02002 

.02043 

29 

30 

.01098 

.01111 

.01371 

.01391 

.01675 

.01703 

.02008 

.02049 

30 

31 

.01103 

.01115 

.01376 

.01395 

.01680 

.01709 

.02013 

.02055 

31 

32 

.01107 

.01119 

.01381 

.01400 

.01685 

.01714 

.02019 

.02061  !  32 

33 

.01111 

.01124  ' 

.01386 

.01405 

.01690 

.01720 

.02025 

.02067 

88 

31 

.01116 

.01128 

.01391 

.01410 

.01696 

.01725 

.02031 

.02073 

34 

35 

.01120 

.01133 

.01398 

.01415 

.01701 

.01731 

.02037 

.02079 

35 

36 

.01124 

.01137 

.01400 

.01120 

.01706 

.01736 

.02042 

.02085 

36 

37 

.01129 

.01142  1 

.01405 

.01425 

.01712 

.01742 

.02048 

.02091 

37 

38 

.01133 

.01146 

.01410 

.01430 

.01717 

.01747 

.02054 

.02097 

38 

39 

.01137 

.01151 

.01415 

.01435 

.01788 

.01753 

.02050 

.02103 

39 

40 

.01142 

.01155 

.01420 

.01440 

.01728 

.01758 

.02066 

.02110 

40 

41 

.01146 

.01160 

.01425 

.01445 

.01733 

.01764 

.02072 

.02116 

41 

42 

.01151 

.01164  ! 

.01430 

.01450 

.01739 

.01769 

.02078 

.02122  i  42 

43 

.01155 

.01169  i 

.01435 

.01455 

.01744 

.01775 

.02084 

.02128 

43 

44 

.01159 

.01173 

.01439 

.01461 

.01750 

.01781 

.02090 

.02134 

44 

45 

.01164 

.01178  i 

.01444 

.01466 

.01755 

.01786 

.02095 

.02140 

45 

48 

.01168 

.01182  i 

.01449 

.01471 

.01760 

.01792 

.02101 

.02146 

46 

4? 

.01173 

.01187  j 

.01454 

•01476 

.01766 

.01798 

.02107 

.02153 

47 

48 

.01177 

.01191 

.01459 

.01481 

.01771 

.01803 

.02113 

.02159 

48 

49 

.01182 

.01196 

.01464 

.01486 

.01777 

.01809 

.02119 

.02165 

49 

50 

.01186 

.01200 

.01469 

.01491 

.017-82 

.01815 

.02125 

.02171 

50 

51 

.01191 

.01205 

.01474 

.01496 

.01788 

.01820 

.02131 

.02178 

51 

52 

.01195 

.01209 

.01479 

.01501 

.01793 

.01826  ; 

.02137 

.02184 

52 

53 

.01200 

.01214 

.01484 

.01506 

.01799 

.01832 

.02143 

.02190 

53 

54 

.01204 

.01219 

.01489 

.01512 

.01804 

.01&37 

.02149 

.02196 

54 

55 

.01209 

.01223 

.01494 

.01517 

.01810 

.01843 

.02155 

.02203 

55 

56 

.01213 

.01228 

.01499 

.01522 

.01815 

.01S49 

.02161 

.02209 

56 

57 

.01218 

.01233 

.01504 

.01527 

.01821 

.01854 

.02167 

.02215   57 

58 

.01222 

.01237 

.01509 

.01532 

.01826 

.01860 

.02173 

.02221   58 

59 

.01227 

.01242 

.01514 

.01537 

.01832 

.0186(5 

.02179 

.02228 

59 

60 

.01231 

.01247 

.01519 

.01543 

.01837 

.01872 

.02185 

.02234 

60 

TABLE  XIII.-VERSINES  AN 


135 


! 
/ 

12° 

13° 

14° 

gALIFOR^^^ 

L$* 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.02185 

.02234  i 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

0 

1 

.02191 

.02240 

.02570 

.02637 

.02977 

.03009 

.03415 

.0353(5 

1 

2 

.02197 

.02247 

.02576 

.02644 

.02985 

.03076 

.03422 

.03544 

2 

3 

.02203 

.02253 

.02583 

.02651 

.02992 

.03084 

.08480 

.03552 

3 

4 

.02210 

.02259 

.02589 

.02658 

.02999 

.03091 

.03438 

.03560 

4 

5 

.02216 

.02206 

.02596 

.02665 

.03006 

.03099 

.03445 

.03568 

5 

6 

.02222 

.02272  ! 

.02602 

.02672 

.03013 

.03106 

.03453 

.03576 

6 

7 

.02228 

.02279 

.02609 

.02679 

.03020 

.03114 

.03460 

.035*1 

7 

8 

.022:34 

.02285 

.02616 

.02686 

.03027 

.03121 

.03468 

.03592 

8 

9 

.02240 

.02291 

.02622 

.02693 

.03034 

.03129 

.03476 

.03601 

9 

10 

.02246 

.02298 

.02629 

.02700 

.03041 

.03137 

.03483 

.03609 

10 

11 

.02252 

.02304 

.02635 

.02707 

.03048 

.03144 

.03491 

.03617 

11 

12 

.02258 

.02311 

.02642 

.02714 

.03055 

.03152 

.03438 

.03625 

12 

13 

.02265 

.02317 

.02649 

.02721 

.03063 

.03159 

.03506 

03633 

13 

14 

.02271 

.02323 

.02655 

.02728 

.03070 

.03167 

.03514 

.03642 

14 

15 

.02277 

.02330 

.02662 

.02735 

.03077 

.03175 

.03521 

.03650 

15 

16 

.02283 

.02336 

.02669 

.02742 

.03084 

.03182 

.03529 

.03658 

16 

17 

.02289 

.02343 

.02675 

.02749 

.03091 

.03190 

.0*537 

.03666 

17 

18 

.02295 

.02319 

.02682 

.02756 

.03098 

.03198  • 

.0:3544 

.03674 

18 

19 

.02302 

.02356 

.02689 

.02763 

.03106 

.03205  • 

.03552 

.03683 

19 

20 

.02308 

.02362 

.,02696 

.02770 

.03113 

.03213 

.03560 

.03691 

20 

21 

.02314 

.02369 

.02702 

.02777 

.03120 

.03221 

.03567 

.03699 

21 

22 

.02320 

.02375 

.02709 

.027?4 

.03127 

.03228 

.03575 

.03708 

22 

23 

.02327 

.02382 

.02716 

.02791  i 

.03134 

.03236 

.03583 

.03716 

23 

24 

.02333 

.02388 

.02722 

.02799  ! 

.03142 

.03244 

.03590 

.03724 

24 

25 

.02339 

.02395 

.02729 

.02806  i 

.03149 

.03251 

.03598 

.03732 

25 

26 

.02345 

.C2402  ! 

.02706 

.02813 

.03156 

.03259 

.03606 

.03741 

26 

27 

.02352 

.02408 

.02743 

.02820  i 

.03163 

.03267 

.03614 

.03749 

27 

28 

.02358 

.02415 

.02749 

.02827  | 

.03171 

.03275 

.03621 

.03758 

28 

29 

.02364 

.02421  ! 

.02756 

.02834  i 

.03178 

.03282 

.03629 

.03766 

29 

30 

.02370 

.02-128  ! 

.02763 

.02842  | 

.03185 

.03290 

.03637 

.03774 

30 

31 

.02377 

.02435 

.02770 

.02849 

.03193 

.03298 

.  03645 

.03783 

31 

32 

.02383 

.02441 

.02777 

.02856 

.03200 

.03306 

.03653 

.03791 

32 

33 

.02389 

.02448 

.02783 

.02803 

.03207 

.03313 

.03600 

.03799 

33 

34 

.02396 

.02454 

.02790 

.02870 

.03214 

.03321 

.03668 

.03808 

34 

35 

.02402 

.02401 

.02797 

.02878 

.03222 

.03329 

.03676 

.03816 

35 

36 

.02408 

.02468 

.02804 

.02885 

.03229 

.03337  ; 

.03684 

.03825 

36 

37 

.02415 

.0247'4 

.02811 

.02892 

.03236 

.03345 

.03692 

.03833 

37 

38 

.02421 

.02481 

.02818 

.02899 

.03244 

.03353 

.03699 

.03842 

38 

39 

.02427 

.02488 

.02824 

.02907 

.03251 

.03360 

.03707 

.03850 

20 

40 

.02434 

.02494 

.02831 

.02914 

.03258 

.03368 

.03715 

.03858 

40 

41 

.02440 

.02501 

.02838 

.02921 

.03266 

.03376 

.03723 

.03867 

41 

42 

.02447 

.02508 

.02845 

.02928 

.03273 

.03384 

.03731 

.03875 

42 

43 

.02453 

.02515 

.02852 

.  02336 

.03281 

.03392 

.03739 

.03884 

43 

44 

.02459 

.02521  1  .02859 

.02943 

.03288 

.03400 

.03747 

.03892 

44 

45 

.02466 

.02528 

.02866 

.02950 

.03295 

.03408 

.03754 

.03901 

45 

46 

.02472 

.02535 

.02873 

.02958 

.03303 

.oaiie 

.03762 

.03909 

46 

47 

.02479 

.02542 

.02880 

.02965 

.03310 

.03424 

.03770 

.03918 

47 

48 

.02485 

.02548 

.02887 

.02972 

.03318 

.03432 

.03778 

.03927 

48 

49 

.02492 

.02555 

.02894 

.02980 

.03325 

.03439 

.03786 

.03935 

49 

50 

.02498 

.02562 

.02900 

.02987 

.03333 

.03447 

.03794 

.03944 

50 

51 

.02504 

.02569 

.02907 

.02994 

.03340 

.03455 

.03802 

.03952 

51 

52 

.02511 

.02576 

.02914 

.03002 

.0:3347 

.03463 

.03810 

.03961 

52 

53 

.02517 

.02582 

.02921 

.0:3009 

.03355 

.03471 

.03818 

.03969 

53 

54 

.02524 

.02589 

.02928 

.03017 

.0,3362 

.03479 

.03826 

.03978 

54 

55 

.02530 

.02596 

.02935 

.03024 

.03370 

.03487 

.03834 

.03987  55 

56 

.02537 

.02603 

.02942 

.03032 

.03377 

.03495 

.03842 

.03995  56 

57 

.02543 

.08610 

.02949 

.03039 

.08385 

03503 

.03850 

.04004  57 

58 

.02550 

.02617 

.02956 

.03046 

i03392 

.03512 

.03858 

.04013  58 

59 

.02556 

.02834 

.02063 

.03054 

.03400 

.03520 

.03866 

.04021 

59 

60 

.02563 

.02630 

.02970 

.0:1001 

.03407 

.03528 

.03874 

.04030 

60 

336 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

1( 

)° 

17 

0 

li 

IS 

JO 

Vers. 

Exsec.  i 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.04030 

.04370 

.04569 

.04894 

.05146 

.05448 

.05762 

0 

1 

.04039  ; 

.04378 

.04578 

.04903 

.05156 

.05458 

.05773 

1 

2 

.04047 

.04387 

.04588 

.04912 

.05166 

.05467 

.05783 

2 

3 

.03898 

.04056 

.04395 

.04597  ' 

.04921 

.05176 

.05477 

.05794 

3 

4 

.03906 

.04065 

.04404 

.04606 

.04930 

.05186 

.05486 

.05805 

4 

5 

.03914 

.04073 

.04412 

.04616 

.04939 

.05196 

.05496 

.05815 

5 

6 

.03922 

.04082 

.04421 

.04025 

.04948 

.05206 

.05505 

.05826 

6 

.03930 

.04091 

.04429 

.04635 

.04957 

.05216 

.05515 

.05836 

7 

8 

.03938 

.04100 

.04438 

.04644 

.04967 

.05226 

.05524 

.05847 

8 

9 

.03946 

.04108 

.04446 

.04653 

.04976 

.05236 

.05534 

.05858 

9 

10 

.03954 

.04117 

.04455 

.04663 

.04985 

.05246 

.05543 

.05869 

10 

11 

.03963 

.04126 

.04464 

.04672 

.04994 

.05256 

.05553 

.05879 

11 

12 

.03971 

.04ia5 

.04472 

.04682 

.05003 

.05266 

.05562 

.05890 

12 

13 

.03979 

.04144 

.04481 

.04691 

.05012 

.05276 

.05572 

.05901 

13 

14 

.03987 

.04152 

.04489 

.04700 

.05021 

.05286 

.05582 

.05911 

14 

15 

.03995 

.04161 

.04498 

.04710 

.05030 

.05297 

.05591 

.05922 

15 

16 

.04003 

.04170 

.04507 

.04719 

.05039 

.05307 

.05601 

.05933 

16 

17 

.04011 

.04179 

.04515 

.04729 

.05048 

.05317 

.05610 

.05944 

17 

18 

.04019 

.04188 

.04524 

.04738 

.05057 

.05327 

.05620 

.05955 

18 

19 

.04028 

.04197 

.04533 

.04748 

.05067 

.05337 

.05630 

.05965 

19 

20 

.04036 

.04206 

.04541 

.04757 

.05076 

.05347 

.05639 

.05976 

20 

21 

.04044 

.04214 

.04550 

.04767 

.05085 

.05357 

.05649 

.05987 

21 

22 

.04052 

.04223 

.04559 

.04776 

.05094 

.05367 

.05658 

.05998 

22 

23 

.04060 

.04232 

.04567 

.04786 

.05103 

.05378 

.05668 

.06009 

23 

24 

.04069 

.04241 

.04576 

.04795 

.05112 

.05388 

.05678 

.06020 

24 

25 

.04077 

.04250 

.04585 

.04805 

.05122 

.05398 

.05687 

.06030 

25 

26 

.04085 

.04259 

.04593 

.04815 

.05131 

.05408 

.05697 

.06041 

26 

27 

.04093 

.04268 

.04602 

.04824 

.05140 

.05418 

.05707 

.06052 

27 

28 

.04102 

.04277 

.04611 

.04834 

.05149 

.05429 

.05716 

.06063 

28 

29 

.04110 

.04286 

.0-1620 

.04843 

.05158 

.05439 

.05726 

.00074 

29 

30 

.04118 

.04295 

.04628 

.04853 

.05168 

.05449  ; 

.05736 

.06085 

30 

31 

.04126 

.04304 

.04637 

.04863 

.05177 

.05460 

.05746 

.06096 

31 

32 

.04135 

.04313 

.04646 

.04872 

.05186 

.05470 

.05755 

.06107 

32 

33 

.04143 

.04322 

.04655 

.04882 

.05195 

.05480 

.05765 

.06118 

33 

34 

.04151 

.04331 

.04663 

.04891 

.05205 

.05490 

.05775 

.06129 

34 

35 

.04159 

.04340 

.04672 

.04901 

.05214 

.05501 

.05785 

.06140 

35 

36 

.04168 

.04349 

.04681 

.04911 

.05223 

.05511 

.05794 

.06151 

36 

37 

.04176 

.04358 

.04690 

.04920 

.05232 

.05521 

.05804 

.06162 

37 

38 

.04184 

.04367 

.04699 

.04930 

.05242 

.05532 

.05814 

.06173 

38 

39 

.04193 

.04376 

.04707 

.04940 

.05251 

.C5542 

.05824 

.06184 

39 

40 

.04201 

.04385 

.04716 

.0495J 

.05260 

.05552 

.05833 

.06195 

40 

41 

.04209 

.04394 

.04725 

.04959 

.05270 

.05563 

.05843 

.06206 

41 

42 

.04218 

.04403 

.04734 

.04969 

.05279 

.05573 

.05853 

.06217 

42 

43 

.04226 

.04413 

.04743 

.04979 

.05288 

.05584 

.05863 

.06228 

43 

44 

.04234 

.04422 

.04752 

.04989 

.05298 

.05594 

.05873 

.00239 

44 

45 

.04243 

.04431 

.04760 

.04998 

.05307 

.05604 

.05882 

.06250 

45 

46 

04251 

.04440 

.04769 

.05008 

.05316 

.05615 

.05892 

.06261 

46 

47 

.04260 

.04449 

.04778 

.05018 

.05326 

.05625 

.05902 

.00272 

47 

48 

.04268 

.04458 

.04787 

.05028 

.05335 

.05636 

.05912 

.00283 

48 

49 

.04270 

.04468 

.04796 

.05038 

.05344 

.05646 

.05923 

.0625)5 

49 

50 

.04285 

.04477 

.04805 

.05047 

.05354 

.05657  : 

.05932 

.06306 

50 

51 

.04293 

.04486 

.04814 

.05057 

.05363 

.05667 

.05942 

.06317 

51 

52 

.04302 

.04495 

.04823 

.05067 

.05373 

.05678 

.05951 

.06328 

52 

53 

.04310 

.04504 

.04832 

.05077 

.05382 

.05688 

.05961 

.06339 

53 

54 

.04319 

.04514 

.04841 

.05087 

.05391 

.05699 

.05971 

.06350 

54 

55 

.04327 

.04523 

.04&50 

.05097 

.05401 

.05709 

.05981 

.06362 

55 

56 

.04336 

.04532 

.04858 

,05107 

.05410 

.05720  i 

.05991 

.06373 

56 

57 

.04344 

.04541 

,04867 

.05116 

.05420 

.05730  : 

.06001 

.06384 

57 

58 

.04:i53 

.04551 

.04876 

.05126 

,05429 

.05741 

.06011' 

.0(5395 

58 

59 

.04361 

.04560  ! 

.04885 

,05138 

,05439 

.05751 

.06021 

.06407 

58 

60 

.04370 

.04509 

.04894 

.05146  i 

.054-18 

,05762 

.06031 

.06418 

60 

TABLE  XIII.-VERSINES  AND   EXSECANTS. 


337 


2< 

)° 

2 

L° 

21 

J° 

2! 

J° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.06031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

0 

1 

.06041 

.06429 

.0(5652 

.07126 

.07293 

.07866 

.07961 

.08649 

1 

2 

.06051 

.06440 

.06663 

.07138 

.07:303 

.07879 

.07972 

.08663 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07-984 

.08676 

3 

4 

.06071 

.06463 

.06684 

.07102  | 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06694 

.07174 

.07336 

.07017 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186  ! 

.07347 

.07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199  i 

.07358 

07943 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

.07369 

.07955 

.08041 

.08744 

8 

9 

.06121 

.06520 

.06736 

.OT<:23 

,  .07380 

.07968 

.08052 

.08757 

9 

10 

.06131 

.06531 

.06747 

.07235 

.07391 

.07981 

.08064 

.08771 

10 

11 

.06141 

.06542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

12 

.06151 

.06554 

.06768 

.07259 

.07413 

.08006 

.08086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.06789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

is 

.06181 

.06588 

.06799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.06600 

.06810 

.07307 

.07457 

.08058 

.08132 

.08852 

16 

17 

.06201 

.06611 

.06820 

.07320 

.07468 

.08071 

.08144 

.08866 

17 

18 

.06211 

.06622 

.06831 

.07332 

.07479 

.08084 

.08155 

.08880 

18 

19 

.06221 

.06634 

.06841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.05231 

.06645 

.06852 

.07356 

.07501 

.08109 

.08178 

.08907 

20 

21 

.06241 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

O-> 

.06252 

.06668 

.06873 

.07380 

.07523 

.08135 

.08201 

,08934 

22 

23 

.06263 

.06680 

.06884 

.07393 

.07534 

.08148 

.08213 

.08948 

23 

24 

.06272 

.0(3691 

.06894 

.07405 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.06703 

.06905 

.07417 

.07556 

.08174  j 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.06726 

.06926 

.07442 

.07579 

..08200  i 

.08259 

.09003 

27 

28 

.06312 

.06738 

.06937 

.07454 

.07590 

.08213  i 

.08271 

.09017 

28 

29 

.06323 

.06749 

.06948 

.07466 

.07601 

.08226  ! 

.08282 

.09030 

29 

30 

.06333 

.06761 

.06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.06773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06353 

.06784 

.06980 

.07.503 

.07634 

.08265 

.08317 

.09072 

32 

88 

.06363 

.06796 

i  .06990 

.07516 

.07(545 

.08278 

.08329 

.09086 

33 

34 

.06374 

.06807  1 

|  .07001 

.07528 

.07657 

.08291 

.08340 

.09099 

3^1 

35 

.06384 

.06819 

!  .07012 

.07540 

.07668 

.08305 

.08352 

.09113 

35 

36 

.06394 

.06831  i 

i  .07022 

.07553 

.07679 

.08318 

.08364 

.09127 

36 

37 

.06404 

.06843 

.07033 

.07565 

.07690 

.08331 

.08375 

.09141 

37 

33 

.06415 

.06854 

.07044 

.07578 

.07701 

.08344 

.08387 

.09155 

38 

39 

.06425 

.06866  ! 

.07055 

.07590  , 

.07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878 

.07065 

.07602 

.07724 

.08370 

.08410 

.09183 

40 

41 

.06445 

.06889  ' 

.07076 

.07615  ! 

.07735 

,08383 

.08422 

.09197 

41 

42 

.06456 

.06901  i 

.0?087 

.07827 

.07746 

.08397  •• 

.08434 

.09211 

42 

43 

.06466 

.06913 

.07098 

.07640  ! 

.07757 

.08410 

.08445 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936  : 

.07119 

.07665  ; 

.07780 

.08436 

.08-469 

.09252 

45 

46 

.06497 

.06948 

.071:30 

.07677 

.07791 

.OS44!) 

.08481 

.09266 

46 

47 

.06507 

.069(50 

.07141 

.07690 

.07802 

.08463  | 

.08492 

.09280 

47 

48 

.06517 

.06972  ; 

.07151 

.07702 

.07814 

.08476 

.08504 

.09294 

48 

49 

.06528 

.06984 

.07162 

.07715 

.07825 

.08489  ! 

.08516 

.09308 

49 

50 

.065:38 

.06995  : 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

50 

51 

.06548 

.07007 

.07184 

.07740 

.07848 

.08516 

.08539 

.09a37 

51 

52 

.06559 

.07019 

.07195 

.07752 

.07859 

.08529  i 

.08551 

.09351 

52 

53 

.06569 

.07031  j 

.07206 

.07765 

.07870 

.08542 

.08563 

.09365 

53 

54 

.06580 

.07043 

.07216 

.07778 

.07881 

.08556 

.08575 

.09379 

54 

55 

.06590 

.07055 

.07227 

.07790 

.07893 

.08569  i 

.08586 

.09393 

55 

56 

.06600 

.070(57 

.072:38 

.07803  j 

.07904 

.08582 

.08598 

.09407 

56 

57 

.06611 

.07079 

.07249 

.07816  | 

.07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.07091 

.07260 

.07828  1 

.07927 

.08(509 

.08622 

.09435 

58 

59 

.OW32 

.07103 

.07271 

.07841 

.07938 

.08623 

.08634 

.09449 

59 

80 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

.08645 

.09464  i 

60 

338 


TABLE   XIII.— VERSINES  AND   EXSECANTS. 


1 

24° 

25° 

26° 

27° 

/ 

Vers. 

Exsec. 

Vcrs. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0   .08645 

.09464 

.09369 

.10338 

.10121 

.11260 

.10899 

.12233 

0 

1 

.08657 

.09478 

.09382 

.10353 

.10133 

.11276 

.10913 

.  12249 

1 

m 

.08669 

.09492 

.09394 

.10368 

.10146 

.11292 

.10926 

.12266 

2 

3 

!  08681 

.09506 

.09406 

.10383 

.10159 

.11308 

.10939 

.12283 

3 

4 

.08693 

.09520 

.09418 

.10398 

.10172 

.11323 

.10952 

.12299 

4 

5 

.08705 

.09535 

.09431 

.10413 

.10184 

.11339 

.10965 

.12316 

5 

6 

.08717 

.09549 

.09443 

.10428 

.10197 

.11355 

.10979 

.12333 

6 

7 

.  08728 

.09563 

.09455 

.10443 

.10210 

.11371 

.10992 

.12349 

7 

8 

.08740 

.09577 

.09468 

.10458 

.  10223 

.11387 

.11005 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473 

.10236 

.11403 

.11019 

.12383 

9 

10 

.08764 

.09606 

.09493 

.10488 

.10248 

.11419 

.11032 

.12400 

10 

11 

.08776 

.09620 

.09505 

.10503 

.10261 

.11435 

.11045 

.12416 

11 

12 

.08788 

.09635 

.09517 

.10518 

.10274 

.11451 

.11058 

.12433 

12 

13 

.08800 

.09649 

.09530 

.  10533 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549 

.  10300 

.11483 

.11085 

.12467 

14 

15 

.08824 

.09678 

.09554 

.10564 

.10313 

.11499 

.11098 

.12484 

15 

16 

.08&36 

.09692 

.09567 

.10579 

.10326 

.11515 

.11112 

.12501 

16 

17 

.08848 

.09707 

.09579 

.  10594 

.10338 

.11531 

.11125 

.12518 

17 

18 

.08860   .09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

18 

19 

.08872 

.09735 

.09604 

.  10625 

.10364 

.11563 

.11152 

.12551 

19 

20 

.08884 

.09750 

.09617 

.10640 

.10377 

.11579 

.11165 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10390 

.11595 

.11178 

.12585 

21 

22 

.08908 

.09779 

.09642 

.10670 

.10403 

.11611 

.11192 

.12602 

22 

23 

.08920 

.09793 

.09654 

.  10686 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701 

.10429 

.11643 

.11218 

.12636 

24 

25 

.08944 

.09822 

.09679 

.10716 

.10442 

.11659 

.11232 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

26 

27 

.08968 

.09851 

.09704 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

28 

.08980 

.09866 

.09716 

.10762 

.10481 

.11708 

.11272 

.12704 

28 

29 

.08992 

.09880 

.097-29 

.10777 

.10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895 

.09741 

.10793 

.10507 

.117'40 

.11299 

.12738 

30 

31 

.09016 

.09909 

.09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

32 

.09028 

.09924 

.09767 

.10824 

.10533 

.11772 

.11326 

.12772 

32 

33 

.09040 

.09939 

.09779 

.  10839 

.10546 

.11789 

.11339 

.12789 

33 

34 

.09052 

.09953 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

35 

.09064 

.09968 

.09804 

.10870 

.10572 

.11821 

.11366 

.12824 

35 

36 

.09076 

.09982 

.09817 

.10885 

.10585 

.11838 

.11380 

.12841 

36 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12&58 

37 

38 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.10932 

.10624 

.11886 

.11420 

.12892 

39 

40 

.09125 

.10041 

.09867 

.10947 

.10637 

.11903 

.11434 

.12910 

40 

41 

.09137 

.10055 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

42 

.09149 

.10071 

.09892 

.10978 

.10663 

.11936 

.11461 

.12944 

42 

43 

.09161 

.10085 

.09905 

.10994 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

.09918 

.11009 

.10689 

.11968 

.11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.  10702 

.11985 

.11501 

.12996 

4,5 

46 

.09198 

.10130 

.09943 

.11041 

.10715 

.12001 

.11515 

.13013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11528 

.13031 

47 

48 

.09222 

.10159 

.09968 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

.09981 

.11087 

.10755 

.12051 

.11555 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11569 

.13083 

50 

51 

.09259 

.10204 

.10006 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

.10019 

.11134 

.10794 

.12100 

.11596 

.13117 

52 

53 

.09283 

.10233 

.10032 

.11150 

.10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.  12133 

.11623 

.13152 

54 

55 

.09308   .10263 

.10057 

.11181 

.10833 

.12150 

.11637 

.13170 

55 

56 

.09320 

.10278 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.09332 

.10293 

.10082 

.11213 

.10860 

.12183 

|  .11664  |  .13205 

57 

58 

.09345 

.10308 

!  10095 

.11229 

.10873 

.12199 

.11(578 

.13222 

58 

69 

.09357 

.10323 

.10108 

.11244 

.10886 

.12216 

j  .11692 

.13210 

59 

60 

.09369   .10338 

.10121 

.11260  ! 

.10899 

.12233 

.11705 

.13257 

60 

TABLE   XIII.-VERSINES  AND   EXSECANTS. 


339 


2 

>° 

2 

3° 

3( 

)° 

3] 

1° 

0 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

~ 

0 

.11705 

.13257 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

1 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.16684 

1 

2 

.11733 

.13292 

.12566 

.14372 

.13427 

.15509 

.14313 

.16704 

2 

8 

.11746 

.13310 

.12580 

.14391 

.13441 

.15528 

.14328 

.16725 

3 

4 

.11760 

.1.3327 

.12595 

.14409 

.13456 

.15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.1.3470 

.  15567 

.14358 

.16766 

5 

6 

.11787 

.13362 

.12623 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12637 

.14465 

.13499 

.15606 

.14388 

.16806 

7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15626 

.14403 

.16827 

8 

9 

.11828 

.13415 

.12(305 

.14502 

.13529 

.15645 

.14418 

.16848 

9 

10 

.11842 

.13433 

.12679 

.14521 

.13543 

.15665 

.14433 

.16868 

10 

11 

.11856 

.1,3451 

.12694 

.14539 

.13558 

.15684 

.14449 

.16889 

11 

12 

.11870 

.13468 

.12708 

.14558 

.13573 

.15704 

.14464 

.16909 

12 

13 

.11883 

.13486 

.12722 

.14576 

.13587 

.15724 

.14479 

.16930 

13 

14 

.11897 

.13504 

.12736 

.14595 

.13602 

.15743 

.14494 

.1(5950 

14 

15 

.11911 

.13521 

.12750 

.14014 

.13616 

.15763 

.14509 

.16971 

15 

16 

.11925 

.13539 

.12765 

.14632 

.13631 

.15782 

.14524 

.16992 

16 

17 

.11938 

.13557 

.1277-9 

.14651 

.13646 

.15802 

.14539 

.17012 

17 

18 

.11952 

.13575 

.12793 

.146?'0 

.13660 

.15822 

.14554 

.17033 

18 

19 

.11966 

.13593 

.12807 

.14689 

.13675 

.15841 

.14569 

.17C54 

19 

20 

.11980 

.13610 

.12822 

.14707 

.13690 

.15861 

.14584 

.17075 

20 

21 

.11994 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

22 

.12007 

.13046 

.12850 

.14745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12864 

.14764 

.13734 

.15920 

.14630 

.17137 

23 

24 

.12035 

.13682 

.12879 

.14782 

.13749 

.15940 

.14645 

.17158 

24 

25 

.12049 

.13700 

.12893 

.14801 

.13763 

.15960 

.14660 

.17178 

25 

26 

.12063 

.13718 

.12907 

.14820 

.13778 

.15980 

.14675 

.17199 

26 

27 

.12077 

.13735 

.12921 

.14839 

.13793 

.16000 

.14690 

.17220 

27 

28 

.12091 

.13753 

.12936 

.14858 

.13808 

.16019 

.14706 

.17241 

28 

29 

.12104 

.13771 

.12950 

.14877 

.13822 

.16039 

.14721 

.17262 

29 

30 

.12118 

.13789 

.12964 

.14896 

.13837 

.16059 

.14736 

.17283 

30 

31 

.12132 

.13807 

.12979 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825 

.12993 

.14933 

.13867 

.16099 

.14766 

.17325 

32 

33 

.12160 

.13843 

.13007 

.1495? 

.13881 

.16119 

.14782 

.17346 

33 

34 

.12174 

.13861 

.13022 

.14971 

.13896 

.16139 

.14797 

.17367 

34 

35 

.12188 

.13879 

.13036 

.14990 

.13911 

.16159 

.14812 

.17388 

35 

36 

.12202 

.13897 

.13051 

.15009 

.13926 

.16179 

.14827 

.17409 

36 

37 

.12216 

.13916 

.13065 

.15028 

.13941 

.16199 

.14843 

.17430 

37 

38 

.12230 

.13934 

.13079 

.15047 

.13955 

.16219 

.14858 

.17451 

38 

39 

.12244 

.13952 

.13094 

.15066 

.13970 

.16239 

.1487-3 

.  747'2 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.16259 

.14888 

.  7493 

40 

41 

.12271 

.13988 

.13122 

.15105 

.14000 

.16279 

.14904 

.  7514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.16299 

.14919 

.  7535 

42 

43 

.12299 

.14024 

.13151 

.15143 

.14030 

.16319 

.14934 

.  7556 

43 

44 

.12313 

.14042 

.13166 

.15162 

.14044 

.16339 

.14949 

.  7577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.14965 

.  7598 

45 

46 

.12341 

.14079 

.13195 

.15200 

.14074 

.16380 

.14980 

.  7620 

46 

47 

.12355 

.14097 

.13209 

.15219 

.14089 

.16400 

.14995 

.  7641 

47 

48 

.12369 

.14115 

.13223 

.15239 

.14104 

.16420 

.15011 

.  7062 

48 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.16440 

.15026 

.  7-683 

49 

50 

.12397 

.14152 

.13252 

.15277 

.14134 

.16460 

.15041 

.  7704 

50 

51 

.12411 

.14170 

.13267 

.15296 

.14149 

.16481 

.15057 

.  7726 

51 

52 

.12425 

.14188 

.13281 

.15315 

.14164 

.16501 

.1507-2 

.  7747 

52 

53 

.12439 

.14207 

.13296 

.15:335 

.14179 

.16521 

.15087 

.  7768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.14194 

.16541 

.  15103 

.  7790 

54 

55 

.12468 

.14243 

.1-3325 

.15373 

.14208 

.  16562 

.15118 

.  7811 

55 

56 

.12482 

.14262 

.13339 

.15393 

.14223 

.16582 

.15134 

.17832 

56 

57 

.12496 

.14280 

.13354 

.15412 

.14238 

.16602 

.15149 

.17854 

57 

58 

.12510 

.14299 

.13368 

.15431 

.14253 

.16623 

.15164 

.17875 

58 

59 

.  12524 

.14317 

.13383 

.15451 

.14268 

.16643 

.15180 

.17896 

59 

60 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

.15195 

.17918 

60 

340 


TABLE  XIII.-VERSINES  AND   EXSECANTS. 


' 

32' 

33° 

34° 

35° 

1 
/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.15195 

.17918 

.16133 

.19230 

.17096 

.20022 

.18085 

.22077 

0 

1 

.15211 

.17939 

.16149 

.1-9259 

.17113 

.20045 

,  18101 

.22102 

1 

2 

.  15226 

.17961 

.10105 

.19281 

.17129 

.20009 

.18118 

.22127 

2 

3 

.15241 

.17982 

.16181 

.19304 

.17145 

.20093 

.18135 

[22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17161 

.20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

.17178 

.20740 

.18168 

.22202 

5 

G 

.15288 

.18047 

.16228 

.19372 

.17194 

.20764 

.18185 

.22227 

6 

7 

.15303 

.18068 

.16244 

.19394 

.17210 

.20r88 

1  .18202 

.22252 

7 

8 

.15319 

.18090 

.16260 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

.15334 

.18111 

.16276 

.19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292 

.19463 

,  .17259 

.20859 

.18252 

.22327 

10 

11 

.15365 

.18155 

.16308 

.194a5 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381 

.18176 

.10324 

.19508 

.17292 

.20907 

.18286 

.22377 

12 

13 

.15396 

.18198 

.16340 

.19531 

.17308 

.20931 

i  .18302 

.22402 

13 

14 

.15412 

.18220 

.16355 

.19554 

j  .17325 

.20955 

.183^9 

.22428 

14 

:  15 

.15427 

.18241 

.16371 

.19576 

i  .17341 

.20979 

.18336 

.22453  15 

16 

.15443 

.18263 

.16387 

.19599 

1  .17357 

.21003 

.18353 

.22478 

16 

17 

.15458 

.18285 

.16403 

.19622 

.17374 

.21027 

.18369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

.17390 

.21051 

.18386 

.22528 

18 

i  19 

.15489 

.18328 

.16435 

.19068 

.17407 

.21075 

.18403 

.22554 

19 

20 

.15505 

.18350 

.16451 

.19691 

.17423 

.21099 

.18420 

.2257'9  i  20 

1  31 

.15520 

.18372 

.10467 

.19713 

.17439 

.21123 

.18437 

.22604  21 

i  22 

.15536 

.18394 

.16483 

.19736 

.17450 

.21147 

.18454 

.22029 

22 

23 

.15552 

.18116 

.16499 

.19759 

.17472 

.21171 

.18470 

.22055 

23 

24 

.15567 

.18437 

.16515 

.19782 

.17489 

.21195 

.18487 

.22080 

24 

25 

.15583 

.18459 

.16531 

.19805 

.17505 

.21220 

.18504 

.227'00  25 

-26 

.15598 

.18481 

.16547 

.19828 

.17582 

.21244 

.18521 

.22731 

26 

27 

.15614 

.18503 

.16563 

.19851 

.17538 

.21208 

[18538 

.22756 

27 

28 

.15630 

.18525 

.16579 

.19874 

.17554 

.21292 

.18555   .227'82 

28 

29 

.15045 

.18547 

.16595 

.19897 

.17571 

.21310 

.18572  i  .22807 

29 

30 

.15001 

.18569 

.16611 

.19920 

.17587 

.21341 

.18588 

.22833 

30 

31 

.15676 

.18591 

.16627 

.19944 

.17604 

.21365 

.18005 

.22858 

31 

32 

.  15692 

.18613 

.16644 

.19907 

.17680 

.21389 

.18022 

.22884 

32 

33 

.15708 

.18635 

.16660  i  .19990 

.17687 

.21414 

.18039 

.22909 

33 

34 

.15723 

.18657 

.16676  !  .20013 

.17658 

.2143,8 

.18056 

.22935 

3-1 

35 

.15739 

.18679 

.16692   .20036 

.17070   .21402 

.18073 

.22960 

35 

36 

.15755 

.18701 

.16708   .20059 

.17USU   .21487 

.18090 

.22986 

36 

37 

.15/70 

.18723 

.16724  i  .20083 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15786 

.18745 

.16740   .20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.15802 

.18707 

.16756  :  .20129 

.17786 

.21500 

.18741 

.23063 

39 

40 

.15818 

.18790 

.16772 

.20152 

.17752 

.21584 

.18758 

.23089 

40 

41 

.15833 

.18812 

.16788 

.20176 

.17769 

.21609 

.18775 

.23114 

41 

42 

45849 

.18834 

.16805 

.20193 

[17786 

[21638 

.18792 

.23140 

42 

43 

.15865 

.18856 

.16821 

.30282 

.17802 

.21658 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837 

[20246 

.17819 

.21082 

.18826 

.23192 

44 

45 

.  15896 

.18901 

.16853 

.20269 

.17835 

.21707 

.18843 

.23217 

45 

46 

.15912 

.18923 

.16809 

.20292 

.17852 

.21731 

.18860 

.23243 

40 

47 

.15923 

.18945 

.16885 

.2C316 

.17808 

.21756 

.18877 

.2320Q 

47 

48 

.15943 

.18967 

.16902 

.20339 

.17885 

.21781 

.18894 

.23295 

48 

49 

.15959 

.18990 

.16918 

•[20363 

[i79oa 

.21805 

.18911 

.23321 

49 

50 

.15975 

.19012 

.16934 

.20386  : 

.17918 

.21830 

48928 

.23347 

50 

51 

.1,-)991 

.19034 

.10950 

.20410 

.17935 

.21855 

.18945 

.23373 

51 

52 

.10006 

.19057 

.16966 

.20133 

.17952 

.21879 

.18962 

.23399 

52 

53 

.1(5022 

.19079 

.16983 

.20157 

.17968 

.21904 

.18979 

.23424 

53 

54 

.16038 

.19102 

.16999 

.20480  !  .17985 

.21929 

.18996 

..23450 

54 

55 

.Io054 

.19124 

.17015 

.2M.-.01  :  .18001 

.21958 

.19013 

.23476 

55 

56 

.16070 

.19146 

.17031 

.30527   .18018 

.21978 

.19030 

.33503 

50 

57 

.16085 

.19169 

.17047 

.L'()",l    .ISO:!.! 

.82008 

.19017   .23529 

57 

58 

.10101 

.19191 

.17064 

.2057T) 

.18051 

.221128 

.19001   .23555 

58 

59 

.16117 

.19214 

.17080 

.89598 

.18068 

.22053 

.19081   .23581 

59 

60 

.16188 

.192.-50 

.17096 

.•JiMtt-  :  .18085 

.22077  il  .19098   .23007 

60 

TABLE  XIIL—  VERSINES  AND  EXSECANTS. 


341 


/ 

36° 

37° 

38° 

39° 

' 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers.  Exsec. 

~0~ 

.19098 

.23607 

.20136 

.25214 

.21199 

.26902 

.22285 

.28876 

0 

1 

.19115 

.23633 

.20154 

.25241 

.21217 

.26931 

.22804 

.28706 

1 

2 

.19133 

.23659 

.20171 

.25269 

.212:35 

.26960 

.22322 

.28737 

2 

3 

.19150 

.23685 

.20189 

.25296 

.21253 

.26988 

.22340 

.28767 

3 

4 

.19167 

.23711 

.20207 

.25324 

.21271 

.27017 

.22359 

.28797 

4 

5 

.1918' 

.23738 

.20224 

.25351  I  .21289 

.27046 

.22377 

.28828 

5 

6 

.19201 

.23764 

.20242 

.25379 

.21307 

.27075 

.22395 

.28858 

6 

7 

.19218 

.23790 

.20259 

.25406 

.21324 

.27104 

.22414 

.28889 

7 

8 

.19235 

.23816 

.20277 

.25434 

.21342 

.27133 

.22432 

.28919 

8 

9 

.19252 

.23843 

.20294 

.25462 

.21360 

.27162 

.2)2450 

.28950 

9 

10 

.19270 

.23869 

.20312 

.25489 

.21378 

.27191 

.22469 

.28980 

10 

11 

.19287 

.23895 

.20329 

.25517 

.21396 

.27221 

.22487 

.29011 

11 

It 

.19304 

.23922 

.20347 

.25545 

.21414 

.27250 

.22506 

.29042 

12 

13 

.19321 

.23948 

.20305 

.25572 

.21432 

.27279 

.22524 

.29072 

13 

14 

.19338 

.23975 

.20382 

.25600 

.21450 

.27308 

.22542 

.29103 

14 

13 

.19356 

.24001 

.20400 

.25628 

.21468 

.27337 

.22561 

.29133 

15 

16 

.19373 

.240^8 

.20417 

.25656 

.21486 

.27366 

.22579 

.29164 

16 

17 

.19390 

.24054 

.20435 

.25683 

.21504 

.27396 

.22598 

.29195 

17 

18 

.19407 

.24081 

.20453 

.25711 

.21522 

.27425 

.22616 

.29226 

18 

19 

.19424 

.24107 

.20470 

.25739 

.21540 

.27454 

.22634 

.29256 

19 

20 

.19442 

.24134 

.20488 

.25767 

.21558 

.27483 

.22653 

.29287 

20 

21 

.19459 

.24160 

.20506 

.25795 

.21576 

.27513 

.22671 

.29318 

21 

t)O 

.19476 

.24187 

.20523 

.25823 

.21593 

.27542 

.22090 

.29349 

22 

23 

.19493 

.24213 

.20541 

.25851 

.21613 

.27572 

.22708 

.29380 

23 

24 

.19511 

.24240 

.20559 

.25879 

.21631 

.27601 

22727 

.29411 

24 

25 

.19528 

.24267 

.20576 

.25907 

.21649 

.27630 

.'22745 

.29442 

25 

26 

.19545 

.24293 

.20594 

.25935 

.21667 

.27660 

.22764 

.29473 

26 

27 

.19562 

.24320 

.20612 

.25963 

.21685 

.27689 

.22782 

.29504 

27 

28 

.19580 

.24347 

.20029 

.25991 

.21703 

.27719 

.22801 

.29535 

28 

29 

.19597 

.24373 

.20647 

.26019  1  .21721 

.27748 

.22819 

.29566 

29 

30 

.19614 

.24400 

.20665 

.26047 

.21739 

.27778 

.22838 

.29597 

30 

31 

.19632 

.24427 

.20682 

.26075 

.21757 

.27807 

.22856 

.29628 

31 

32 

.19649 

.24454 

.20700 

.20104 

.21775 

.27837 

.22875 

.29659 

32 

33 

.19666 

.24481 

.20718 

.26132  | 

.21794 

.27'807 

.22893 

.29690 

33 

34 

.Ia684 

.24508 

.20736 

.26160 

.21812 

.27896 

.22912 

.29721 

34 

.19701 

.24534 

.20753 

.26188 

.21830 

.27926 

.22330 

.29752 

35 

36 

.19718 

.24561 

.20771 

.26216 

.21848 

.27956 

.22949 

.29784 

36 

37 

.19736 

.24588 

.20789 

.26245 

.21866 

.27985 

.22967 

.29815 

37 

38 

.19753 

.24615 

.20807 

.26273 

.21884 

.28015 

.22986 

.29846 

38 

39 

.19770 

.24642 

.20824 

.26301 

.21902 

.28045 

.23004 

.29877 

39 

40 

.19788 

.24669 

.20842 

.26330 

.21921 

.28075 

.23023 

.29909 

40 

41 

.19805 

.24696 

.20860 

.26358 

.21939 

.28105 

.23041 

.29940 

41 

42 

.19822 

.24723 

.20878 

.26387 

.21!)5r 

.28134 

.23000 

.29971 

42 

43 

.19840 

.24750 

.20895 

.26415 

.21975 

.28164 

.23079 

.30003 

43 

44 

.19857 

.24777 

.20913 

.26443 

.21993 

.28194 

.23097 

.30034 

44 

45 

.198T5 

.24804 

.20931 

.26472 

.22012 

.28224 

.23116 

.30066 

45 

46 

.19892 

.24832 

.20949 

.26500 

.22030 

.28254 

.23134 

.30097 

47 

.19909 

.21S59 

.20967 

.26529 

.22048 

.28284 

.23153 

.30129 

47 

48 

.19927 

.24886 

.20985 

.  26557 

.22006 

.28314 

.23172 

.30160 

40 

49 

.19941 

.2(013 

.21002 

.26586 

.22084 

.28344 

.23190 

.30192 

49 

50 

.19962 

.21940 

.21020 

.26615 

.22103 

.28374 

.23209 

.30223 

50 

51 

.19979 

.24967 

.21038 

.20643 

.22121 

.28404 

.23228 

.30255 

51 

52 

.19997 

.24995 

.21056 

.20072 

.22139 

.28434 

.23246 

.30287 

52 

53 

.20014 

.25022 

.21074 

.26701 

.22157 

.28404  ' 

.23265 

.30318 

53 

54 

.20032 

.25049 

.21092 

.26729 

.22176 

.28495 

.23283 

.30350 

54 

55 

.20049 

.25077 

.21109 

.26758 

.22194 

.28525 

.23302 

.30382 

55 

56 

.20066 

.25104 

.21127 

.26787 

!  22212 

.28555 

.23321 

.30413 

56 

57 

.20084 

.25131 

.21145 

.26815 

.23231 

.28585 

.23-339 

.30445 

57 

58 

.20101 

.25159 

.21163 

.20844 

.22249 

.28015 

.23858 

.30477 

53 

59 

.20119 

.25186 

.21181 

.26873 

.22207 

.28646 

.23377 

.30509 

59 

60 

.20136 

.25214 

.21199 

.20902 

.22285 

.28070 

.23396 

.30541 

60 

TABLE  xm.-vetisiNEs  AND  EXSECANTS. 


/ 

40° 

41° 

42° 

43° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.23396 

.30541 

.24529 

.32501 

.25686 

.34563 

.20865 

.30733 

0 

1 

.23414 

.30573 

.24548 

.32535 

.25705 

.34599 

.26884 

.36770 

1 

2 

.23433 

.30605 

.24567 

.32568 

.25724 

.34634 

.26904 

.36807 

2 

3 

.23452 

.30636 

.24586 

.32602 

.25744 

.34069 

.26924 

.36844 

3 

4 

.23470 

.30668 

.24605 

.32636 

.25763 

.34704 

.26944 

.36881 

4 

5 

.23489 

.30700 

.24625 

.32669 

.25783 

.347'40 

.26964 

.36919 

5 

6 

.23508 

.30732 

.24644 

.32703 

.25802 

.34775 

.26984 

.36956 

6 

7 

.23527 

.30764 

.24663 

.32737 

.25822 

.31811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

.32770 

.25841 

.34846 

.27024 

.37030 

8 

9 

.23564 

.30829 

.24701 

.32804 

."25801 

.34882 

.27'043 

.37068 

g 

10 

.23583 

.30861 

.24720 

.32838 

.25880 

.34917 

.27003 

.37105 

10 

11 

.23602 

.30893 

.24739 

.32872 

.25900 

.34953 

.27083 

.37143 

11 

12 

.23020 

.30925 

.24759 

.32905 

.25920 

.34988 

.27103 

.37180 

12 

13 

.23639 

.30957 

.24778 

.32939 

.25989 

.35024 

.27123 

.37218 

13 

14 

.23658 

.30989 

.24797 

.32973 

.25959 

.35060 

.27143 

.37255 

14 

15 

.23677 

.31022 

.24816 

.33007 

.25978 

.35095 

.27103 

.37293 

15 

16 

.23696 

.31054 

.24835 

.33041 

.25998 

.35131 

.27183   .37330 

16 

17 

.23714 

.31086 

.24854 

.33075 

.26017 

.35167 

.27203 

.37368 

17 

18 

.23733 

.31119 

.24874 

.33109 

.26037 

.35203 

.27223 

.37406 

18 

19 

.23752 

.31151 

.24893 

.33143 

.26056 

.35238 

.27243 

.37443 

19 

20 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20 

21 

.23790 

.31216 

.24931 

.33211 

.26096 

.35310 

.27283 

.37519 

21 

22 

.23808 

.31248 

.24950 

.33245 

.26115 

.35346 

.27303 

.37556 

22 

23 

.23827 

.31281 

.24970 

.33279 

.26135 

.35382 

.27323 

.37594 

23 

24 

.23846 

.31313 

.24989 

.33314 

.26154 

.35418 

.27343 

.37632 

24 

95 

.23865 

.31346 

.25008 

,  .33348 

.26174 

.35454 

.27363 

.37670 

25 

26 

.23884 

.31378 

.25027 

.33382 

.26194 

.35-190 

.27383 

.37708 

26 

27 

.23903 

.31411 

.25047 

.3,3416 

.26213 

.35526 

.27403 

.37746 

27 

28 

.23922 

.314-13 

.25066 

.33451 

.20233 

.35562 

.27423 

.37784 

28 

29 

.23941 

.31476 

.25085 

.33485 

.26253 

.35598 

.27443 

.37822 

29 

30 

.23959 

.31509 

.25104 

.33519 

.26272 

.35634 

.27403 

.37860 

30 

31 

.23978 

.31541 

.25124 

.33554 

.26292 

.35670 

.27483 

.37898 

31 

32 

.23997 

.31574 

.25143 

.83588 

.26312 

.35707 

.27503 

.37936 

32 

33 

.24016 

.31607 

.25162 

.33622 

.26331 

.35743 

.27523 

.37974 

33 

34 

.24035 

.31640 

.25182 

.33657 

.26351 

.35779 

.27543 

.38012 

34 

35 

.24054 

.31672 

.25201 

.33691 

.26371 

.35815 

.27563 

.38051 

35 

36 

.24073 

.31705 

.25220 

.33726 

.26390 

.35852 

.27583 

.38089 

36 

37 

.24092 

.31738 

.25240 

.33760 

.26410 

.35888 

.27603 

.38127 

37 

38 

.24111 

.31771 

.25259 

.33795 

.36430 

.35924 

.27023 

.38165 

38 

39 

.24130 

.31804 

.25278 

.33830 

.26449 

.35961 

.27043 

.38204 

39 

40 

.24149 

.31837 

.25297 

.33864 

.26469 

.35997 

.27063 

.38242 

40 

41 

.24168 

.31870 

.25317 

.33899 

.26489 

.36034 

.27083 

.38280 

41 

42 

.24187 

.31903 

.25336 

.339:54 

.26509 

.36070 

.277'03 

.38319 

42 

43 

.24206 

.31936 

.25356 

.33968 

.26528   .36107 

.27723 

.38357 

43 

44 

.24225 

.31969 

.25375 

.34003 

.26548 

.36143 

.27743 

.38396 

44 

45 

.24244 

.32002 

.25394 

.34038 

.26508 

.36180 

.27764 

.38434 

45 

46 

.24262 

.32035 

.25414 

.34073 

.26588 

.36217 

.27784 

.38473 

46 

47 

.24281 

.32068 

.25433 

.34108 

.26607 

.36253 

.27'804 

.38512 

47 

48 

.24300 

.32101 

.25452 

.34142 

.26627 

.36290 

.27824 

.38550 

48 

49 

.24320 

.32134 

.25472 

.34177 

.26647 

.36327 

.27844 

.38589 

49 

50 

.24339 

.32168 

.25491 

.34212 

.26667 

.36363 

.27864 

.38628 

50 

51 

.24358 

.32201 

.25511 

.34247 

.26686 

.36400 

.27884 

.38666 

51 

52 

.24377 

.32234 

.25530 

.34282 

.26706 

.36437 

.27905 

.38705 

UMB 

53 

.24396 

.32267 

.25549 

.34317 

.26726 

.36474 

.27925 

.38744 

53 

54 

.24415 

.32301 

.25569 

.84353 

.267'46 

.36511 

.27945 

.38783 

54 

55 

.24434 

.32334 

.25588 

.84387 

.26766 

.30548 

.27965 

.38822 

55 

56 

.24453 

.32368 

.25608 

.34423 

.26785 

.36585 

.27985 

.38860 

56 

57 

.24472 

.32401 

.25627 

.34458 

.26805 

.36622 

.28005 

.38899 

57 

58 

.24491 

.32434 

.25647 

.34493 

.26825 

.36659 

.28026 

.38938 

58 

59 

.24510 

.32468 

.25066 

.34528 

.26845 

.36696 

.28046 

.38977 

59 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733  | 

.28066 

.39016 

60 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


343 


4< 

1° 

4 

5° 

4 

3° 

4 

7° 

i 

1 

Vers. 

Exsec. 

Vers. 

Exsec.  , 

Vers. 

Exsec. 

Vers. 

Exsec. 

~ 

o 

.28066 

.39016 

.29289 

.41421 

.30534 

.43956  , 

.31800 

.46638 

1 

.28086 

.39055 

.29310 

.41463 

.30555 

.43999 

.31821 

.46674 

1 

2 

.28106 

.39095 

.29330 

.41504  1 

.30576 

.44042 

.31843 

.46719 

2 

3 

.28127 

.39134 

.29351 

.41545 

.30597 

.44086  ! 

.31864 

.46765 

3 

4 

.28147 

.39173 

.29372 

.41586 

.30618 

.44129  : 

.31885 

.46811 

4 

5 

.28167 

.39212 

.29392 

.41627 

.30639 

.44173 

.31907 

.46857 

5 

6 

.28187 

.39251 

.29413 

.41669 

.30660 

.44217  | 

.31928 

.46903 

6 

7 

.28208 

.39291 

.29433 

.41710 

.30681 

.44260 

.31949 

.46949 

7 

8 

.28228 

.39330 

.29454 

.41752 

.30702 

.44304  t 

.31971 

.46995 

8 

9 

.28248 

.39369 

.29475 

,41793 

.30723 

.44347  i 

.31992 

.47041 

9 

10 

.28268 

.39409 

.29495 

.41835 

.30744 

.44391 

.32013 

.47087 

10 

11 

.28289 

.39448 

.29516 

.41876 

.30765 

.44435 

.32035 

.47134 

11 

12 

.28:309 

.39487 

.29537 

.41918 

.30786 

.44479 

.32056 

.47180 

12 

13 

.28329 

.39527 

.29557 

.41959 

.30807 

.44523 

.32077 

.47226 

13 

14 

.28350 

.39566 

.29578 

.42001 

.30828 

.44567 

.32099 

.47272 

14 

15 

.28370 

.39606 

.29599 

.42042 

.30849 

.44610 

.32120 

!  4731  9 

15 

16 

.28390 

.39646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47365 

16 

17 

.28410 

.39685 

.29640 

.42126 

.30891 

.44698 

.32163 

.47411 

17 

18 

.28431 

.39725 

.29661 

.42168 

.30912 

.44742 

.32184 

.47458 

18 

19 

.28451 

.39764 

.29681 

.42210 

.30933 

.44787 

.32205 

.47504 

19 

20 

.28471 

.39804 

.29702 

.42251 

.30954 

.44831 

.32227 

.47551 

20 

21 

.28493 

.39844 

.29723 

.42293 

.30975 

.44875 

.32248 

.47598 

21 

22 

.28512 

.39884 

.29743 

.42335 

.30996 

.44919 

.32270 

.47644 

22 

23 

.28532 

.39924 

.29764 

.42377 

.31017 

.44983 

.32291 

.47691 

23 

24 

.28553 

.39963 

.29785 

.42419 

.31038 

.45007 

.32312 

.47738 

24 

25 

.28573 

.40003 

.29805 

.42461 

.31059 

.45052 

.32334 

.47784 

25 

26 

.28593 

.40043 

.29826 

.42503 

.31030 

.45096 

.32355 

.47831 

26 

27 

.28614 

.40083 

.29847 

.42545 

.31101 

.45141 

.32377 

.47878 

27 

28 

.28634 

.40123 

.29868 

.42587 

.31122 

.45185 

.32398 

.47925 

28 

29 

.28655 

.40163 

.29888 

.42630 

.31143 

.45229 

.32420 

.47972 

29 

30 

.28675 

.40203 

.29909 

.42672 

.31165 

.45274 

.32441 

.48019 

30 

31 

.28695 

.40243 

.29930 

.42714 

.31186 

.45319 

.32462 

.48066 

31 

32 

.28716 

.40283 

.29951 

.42756 

.31207 

.45363 

.32484 

.48113 

32 

33 

.28736 

.40324 

.29971 

.42799 

.31228 

.45408 

.32505 

.48160 

33 

34 

.28757 

.40364 

.29993 

.42841 

.31249 

.45452 

|  .32527 

.48207 

34 

35 

.28777 

.40404 

.30013 

.42883 

.31270 

.45497 

.32548 

.48254 

35 

36 

.28797 

.40444 

.30034 

.42926 

.31291 

.45542 

.32570 

.48301 

36 

37 

.28818 

.40485 

.30054 

.42968 

,  .31312 

.45587 

.32591 

.48349 

37 

38 

.28838 

.40525 

.30075 

.43011 

.31334 

.45631 

.32613 

.48396 

38 

39 

.28859 

.40565 

.30096 

.43053 

.31355 

.45676 

.32634 

.48443 

39 

40 

.28879 

.40606 

.30117 

.43096 

.31376 

.45721 

.32656 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31397 

.45766 

.32677 

.48538 

41 

42 

.28920 

.40687 

.30158 

.43181 

.31418 

.45811 

.32699 

.48586 

42 

43 

.28941 

.40727 

.30179 

.43224 

.31439 

.45856 

.32720 

.48633 

43 

44 

.28961 

.40768 

.30200 

.43267 

.31461 

.45901 

.32742 

.48681 

44 

45 

.28981 

.40808 

.30221 

.43310 

.31482 

.45946 

.32763 

.48728 

45 

46 

.29002 

.40849 

.30242 

.43352 

.31503 

.45992 

.32785 

.48776 

46 

47 

.29023 

.40890 

.30263 

.43395 

.31524 

.46037 

.32806 

.48824 

47 

48 

.29043 

.40930 

.30283 

.43438 

.31545 

.46082 

.32828 

.48871 

48 

49 

.29063 

.40971 

.30304 

.43481 

.31567 

.46127 

.32849 

.48919 

49 

50 

.29084 

.41012 

.30325 

.43524 

.31588 

.46173 

.32871 

.48967 

50 

51 

.29104 

.41053 

.30346 

.43567 

.31609 

.46218 

.32893 

.49015 

51 

52 

.29125 

.41093 

.30367 

.43610 

.31630 

.46263 

.32914 

.490153 

52 

53 

.29145 

.41134 

.30388 

.43653 

.31651 

.46309 

.32936 

.49111 

53 

54 

.29166 

.41175 

.30409 

.43696 

.31673 

.46:354 

.32957 

.49159 

54 

55 

.29187 

.41216 

.30430 

.43739 

.31694 

.46400 

.32979 

.49207 

55 

56 

.29207 

.41257 

.30451 

.43783 

.31715 

.46445 

.33001 

.49255 

56 

57 

.29228 

.41298 

.30471 

.43826 

.31736 

.46491 

.33022 

.49303 

57 

58 

.29248 

.415139 

.30492 

.43869 

.31758 

.4(5537 

.33044 

.49351 

58 

59 

.29269 

.41380 

.30513 

.43912 

.31779 

.46582 

i  .33065 

.49399 

59 

60 

.29289 

.41421 

.30534 

.43956 

.31800 

.46628 

i  .33087 

.49448 

60 

344 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


f 

48° 

49° 

50° 

51" 

f 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.33087 

.49448" 

.34394 

.52425 

.35721 

.55572 

.37068 

.58902 

0 

1 

.33109 

.49496 

.34416 

.52476 

.35744 

.55626 

.37091 

.58959 

1 

2 

.33130 

.49544 

.34438 

.52527 

.35766 

.55680 

.37113 

.59016 

2 

3 

.33152 

.49593 

.34460 

.52579 

.35788 

.557:34 

.37136 

.59073 

3 

4 

.33173 

.49641 

.34482 

.52630 

.35810 

.55789 

.37158 

.59130 

4 

5 

.33195 

.49690 

.34504 

.52681 

.35833 

.55843 

.37181 

.59188 

5 

6 

.33217 

.49738 

.34526 

.52732 

.35855 

.55897 

.37204 

.59245 

6 

7 

.33238 

.49787 

.34548 

.52784 

.35877 

.55951 

.37226 

.59302 

7 

8 

.33260 

.49835 

.34570 

.52835 

.35900 

.56005 

.37249 

.59360 

8 

9 

.33282 

.49884 

.34592 

.52886 

.35922 

.56060 

.37272 

.59418 

9 

10 

.33303 

.49933 

.34614 

.52938 

.35944' 

.56114 

.37294 

.59475 

10 

11 

.33325 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12 

.33347 

.50030 

.34658 

.53041 

.35389 

.56223 

.37340 

.59590 

12 

13 

.33368 

.50079 

.34680 

.53092 

.36011 

.56278 

.37362 

.59648 

13 

14 

.33390 

.50128 

.34702 

.53144 

.38034 

.56332 

.37385 

.59706 

14 

15 

.33412 

.50177 

.34724 

.53196 

.36056 

.56387 

.37408 

.59764 

15 

16 

.33434 

.50226 

.34746 

.53247 

.36078 

.56442 

.37430 

.59822 

16 

17 

.33455 

.50275 

.34768 

.53299 

.36101 

.56497 

.37453 

.59880 

17 

18 

.33477 

.50324 

.34790 

.53351 

.36123 

.56551 

.37476 

.59938 

18 

19 

.33499 

.50373 

.34812 

.53403 

.36146 

.56606 

.37498 

.59996 

19 

20 

.33520 

.50422 

.34834 

.53455 

.36168 

.56661 

.37521 

.60054 

20 

21 

.33542 

.50471 

.34856 

.53507 

.36190 

.56716 

.87644 

.60112 

21 

22 

.33564 

.50521 

.34878 

.53559 

.36213 

.56771 

.37567 

.60171 

22 

23 

.33586 

.50570 

.34900 

.53611 

.36235 

.56826 

.37589 

.60229 

23 

24 

.33607 

.50619 

.34923 

.53663 

.36258 

.56881 

.37612 

.60287 

24 

25 

.33629 

.50669 

.34945 

.53715 

.36280 

.56937 

.37635 

.60346 

25 

26 

.33651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.60404 

26 

27 

.33673 

.50767 

.34989 

.53820 

.36325 

.57047 

.37680 

.60463 

27 

28 

.33694 

.50817 

.35011 

.53872 

.36347 

.57103 

.37703 

.60521 

28 

29 

.33716 

.50866 

.3501)3 

.53924 

.36370 

.57158 

.37726 

.60580 

29 

30 

.33738 

.50916 

.35055 

.53977 

.36392 

.57213 

.37749 

.60639 

30 

31 

.33760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37771 

.60698 

31 

32 

.33782 

.51015 

.35099 

.54082 

.36437 

.57324 

.37794 

.60756 

32 

33 

.33803 

.51065 

.35122 

.54134 

.36460 

.57380 

.37817 

.60815 

33 

34 

.33825 

.51115 

.35144 

.54187 

.36482 

.57436 

.37840 

.6087'4 

34 

35 

.33847 

.51165 

.35166 

.54240 

.36504 

.57491 

.37862 

.60933 

35 

36 

.33869 

.51215 

.35188 

.54292 

.36527 

.57547 

.37885 

.60992 

36 

37 

.33891 

.51265 

.35210 

.54345 

.36549 

.57603 

.37908 

.61051 

37 

38 

.33912 

.51314 

.35232 

.54398  ' 

.36572 

.57659 

.37931 

.61111 

38 

39 

.33934 

.51364 

.35254 

.54451 

.36594 

.57715 

.37951 

.61170 

39 

40 

.33956 

.51415 

.35277 

.54504 

.36617 

.57771 

.37976 

.61229 

40 

41 

.33978 

.51465 

.35299 

.54557 

.36639 

.57827 

.37999 

.61288 

41 

42 

.34000 

.51515 

.35321 

.54010 

.36662 

.57883 

.38022 

.61348 

42 

43 

.34022 

.51565 

.35343 

.54663 

.36684 

.57939 

.38015 

.61407 

43 

44 

.34044 

.51615 

.35365 

.54716 

.36707 

.57995 

.38068 

.61467 

44 

45 

.34065 

.51665 

.35388 

.54769 

.36729 

.58051 

.38091 

.61526 

45 

46 

.34087 

.51716 

.35410 

.54822 

.36752 

.58108 

.38113 

.61586 

46 

47 

.34109 

.51766 

.35132 

.54876 

.36775 

.58164 

.38136 

.61646 

47 

48 

.34131 

.51817 

.35454 

.54929 

.36797 

.58221 

.38159 

.61705 

48 

49 

.34153 

.51867 

185476 

.54982 

.36820 

.58277 

.38182 

.61765 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333 

.38205 

.61825 

50 

51 

.34197 

.51968 

.35521 

.55089 

.36865 

.58390 

.38228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.55143 

.36887 

.58447 

.38251 

.61945 

52 

53 

.34241 

.52069 

.35565 

.55196 

.36910 

.58503 

.38274 

.62005 

53 

54 

.34262 

.52120 

.35588 

.55250 

.36932 

.58560 

.38296 

.62065 

54 

55 

.34284 

.52171 

.35610 

.55303 

.36955 

.58617 

.38319 

.62125 

55 

56 

.34306 

.52222 

.35632 

.55357 

.36978 

.58674 

.38342 

.62185 

56 

57 

.34328 

.52273 

.35654 

.55411 

.37000 

.58731 

.38365 

.62246 

57 

58 

.34350 

.52323 

.35677 

.55465 

.37023 

.58788 

138388 

.62306 

58 

59 

.34372 

.52374 

.35699 

.5.-,;-)  18. 

.37045 

.58845 

.38411 

.62366 

59 

60   .34394 

.52425 

.3oT21 

.55572 

.  .37068 

.58902 

.38434 

.62427 

6Q 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


34; 


51 

3° 

5 

3° 

5< 

5, 

)a 

Yers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.88434 

.62427 

.39819 

.66164  j 

.41221 

.70130  : 

.42642 

.74345 

0 

1 

.38457 

.62487 

.39842 

.66228 

.41245 

.70198  | 

.42666 

.74417 

1 

2 

.38480 

.62548 

.39865 

.66292 

.41269 

.70267 

.42690 

.74490 

2 

3 

.38503 

.62609 

.39888 

.66357 

.41292 

.70335 

.42714 

.74562 

3 

4 

.38526 

.62669 

.39911 

.66421 

.41316 

.70403 

.42738 

.74635 

4 

5 

.38549 

.62730 

.39935 

.66486 

.41339 

.70472 

.42762 

.74708 

5 

6 

.38571 

.62791 

.39958 

.66550 

.41363 

.70540 

.42785 

.74781 

6 

.38594 

.62852 

.39981 

.66615 

.41:386 

.70609 

.42809 

.74854 

7 

8 

.38617 

.62913 

.40005 

.66679 

.41410 

.70677 

.42833 

.74927 

8 

9 

.38640 

.62974 

.40028 

.66744 

.41433 

.70746 

.42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.66809 

.41457 

.70815 

.42881 

.75073 

10 

11 

.38686 

.63096 

.40074 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.63157 

.40098 

.66938 

.41504 

.70953 

.42929 

.75219 

12 

13 

.38732 

.63218 

.40121 

.67003 

.41528 

.71022 

.42953 

.75293 

13 

14 

.38755 

.63279 

.40144 

.67068 

.41551 

.71091 

.42976 

.75366 

14 

15 

.38778 

.63341 

.40168 

.67133 

.41575 

.71160 

.43000 

.75440 

15 

16 

.38801 

.63402 

.40191 

.67199 

.41599 

.71229 

.43024 

.75513 

16 

17 

.38824 

.63464 

.40214 

.67264 

i  .41622 

.71298 

.43048 

.75587 

17 

18 

.38847 

.63525 

.40237 

.67329 

.41646 

.71368 

.43072 

.75661 

18 

19 

.38870 

.63587 

.40261 

.67394 

.41670 

.71437 

.43096 

.75734 

19 

20 

.38893 

.63648 

.40284 

.67460 

.41693 

.71506 

.43120 

.75808 

20 

21 

.38916 

.63710 

.40307 

.67525 

.41717 

.71576 

.43144 

.75882 

21 

22 

.38939 

.63772 

.40331 

.67591 

.41740 

.71646  • 

.43168 

.75956 

22 

23 

.38962 

.63834 

.40354 

.67656 

.41764 

.71715 

.43192 

.76031 

23 

24 

.38985 

.63895 

.40378 

.67722 

.41788 

.71785  i 

.43216 

.76105 

24 

25 

.39009 

.63957 

.40401 

.67788 

.41811 

.71855 

.43240 

.76179 

25 

26 

.39032 

.64019 

.40424 

.67853 

.41885 

.71925 

.43264 

.76253 

26 

27 

.39055 

.64081 

.40448 

.67919 

.41859 

.71995 

.43287 

.76328 

27 

28 

.39078 

.64144 

.40471 

.67985 

.41882 

.72065 

.43311 

.76402 

28 

29 

.39101 

.64206 

.40494 

.68051 

.41906 

.72135 

.43335 

.76477 

29 

3(3 

.39124 

.64268 

.40518 

.68117 

.41930 

.72205 

.43359 

.76552 

30 

31 

.39147 

.64330 

.40541 

.68183 

.41953 

.72275 

.43383 

.76626 

31 

32 

.39170 

.64393 

.40565 

.68250 

.41977 

.72346  I 

.43407 

.76701 

32 

33 

.39193 

.64455 

.40588 

.68316 

.42001 

.72416  ! 

.43431 

.76776 

33 

34 

.39216 

.64518 

.40611 

.68382 

.42024 

.72487 

.43455 

.76851 

34 

35 

.39239 

.64580 

.40635 

.68449 

.42048 

.72557 

.43479 

.76926 

35 

36 

.39262 

.64643 

.40658 

.68515 

.42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

.42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.64768 

.40705 

.68648 

.42119 

.72769 

.43551 

.77152 

38 

39 

.39332 

.64831 

.40728 

.68715 

.42143 

.72840 

.43575 

.77227 

39 

40 

.39355 

.64894 

.40752 

.68782 

.42167 

.72911  ; 

.43599 

.77303 

40 

41 

.39378 

.64957 

.40775 

.68848 

.42191 

.72982  ! 

.43623 

.77378 

41 

42 

.39401 

.65020 

.40799 

.68915 

.42214 

.73053 

.43647 

.77454 

42 

43 

.39424 

.65083 

.40822 

.68982 

.42238 

.73124 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40869 

.69116 

.42285 

.73267  i 

.43720 

.77681 

45 

46 

.39494 

.65272 

.40893 

.69183 

.42309 

.73338 

.43744 

.77757 

46 

47 

.39517 

.65336 

.40916 

.69250 

.42333 

.73409 

.43768 

.77833 

47 

48 

.39540 

.65399 

.40939 

.69318 

.42357 

.73481 

.43792 

.77910 

48 

49 

.39563 

.65402 

.40963 

.69385 

.42381 

.73552 

.43816 

.77986 

49 

50 

.39586 

.65526 

.40986 

.69452 

.42404 

.73624 

.43840 

.78062 

50 

51 

.39610 

.65589 

.41010 

.69520 

.42428 

.73696 

.43864 

.78138 

51 

52 

.39633 

.65653 

.41033 

.69587 

.42452 

.73768 

.43888 

.78215 

52 

53 

.39656 

.65717 

.41057 

.69655 

.42476 

.73840 

.43912 

.78291 

53 

54 

.39679 

.65780 

.41080 

.69723 

.42499 

.73911 

.43936 

.78368 

54 

55 

.39702 

.65844 

.41104 

.69790 

.42523 

.73983 

.43960 

.78445 

55 

56 

.39726 

.65908 

.41127 

.69858 

.42547 

74056 

.43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128 

.44008 

.78598 

57 

58 

.39772 

.66036 

.41174 

.69994 

.4'?595 

.74200 

.44032 

.78675 

58 

59 

.39795 

.66100 

.41198 

.70062 

.48616 

.74272 

.44057 

.78752 

59 

60 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

.44081 

.78829 

CO 

346 


TABLE  Xtll.-VERSINES  AND  EXSECANTS. 


5 

6° 

5 

r° 

5 

jo 

5< 

J° 

f 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.44081 

.78829 

.45536 

.83608 

i  .47008 

.88708 

.48496 

.94160 

0 

1 

.44105 

.78906 

.45560 

.83690 

.47033 

.S8796 

.48521 

.94254 

1 

2 

.44129 

.78984 

.45585 

.83773 

.47057 

.88884 

.48546 

.94349 

2 

3 

.44153 

.79061 

.45609 

.83855 

i  .47082 

.88972 

i  .48571 

.94443 

3 

4 

.44177 

.79138 

.45634 

.83938 

.47107 

.89060 

.48596 

.94537 

4 

5 

.44201 

.79216 

.45658 

.84020 

.47131 

.89148 

.48621 

.94632 

5 

6 

.44225 

.79293 

.45683 

.84103 

.47156 

.89237 

.48646 

.94726 

6 

7 

.44250 

.79371 

.45707 

.84186 

.47181 

.89325 

.48671 

.94821 

7 

8 

.44271 

.79449 

.45731 

.84269 

.47206 

.89414 

.48696 

.94916 

8 

9 

.44293 

.79527 

.45756 

.84352 

.47230 

.89503 

.48721 

.95011 

9 

10 

.44322 

.79604 

.45780 

.84435 

.47255 

.89591 

.48746 

.95106 

10 

11 

.44346 

.79682 

.45805 

.84518 

.47280 

.89680 

.48771 

.95201 

11 

12 

.44370 

.79761 

.45829 

.84601 

.47304 

.89769 

.48796 

.95296 

12 

13 

.44395 

.79839 

.45854 

.84685 

.47329 

.89858 

.48821 

.95392 

13 

14 

.44419 

.79917 

.45878 

.84768 

.47354 

.89948 

.48846 

.95487 

14 

15 

.44443 

.79995 

.45903 

.84852 

.47379 

.90037 

.48871 

.95583 

15 

16 

.44467 

.80074 

.45927 

.84935 

.47403 

.90126 

.48896 

.95678 

16 

17 

.44491 

.80152 

.45951 

.85019 

.47428 

.90216 

.48921 

.95774 

17 

18 

.44516 

.80231 

.45976 

.85103 

.47453 

.90305 

.48946 

.95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

.47478 

.90395 

.48971 

.95966 

19 

20 

.44564 

.80388 

.46025 

.85271 

.47502 

.90485 

.48996 

.96062 

20 

21 

.44588 

.80467 

.46049 

.85355 

.47527 

.90575 

.49021 

.96158 

21 

22 

.44612 

.80546 

.46074 

.85439 

.47552 

.90665 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85523 

.47577 

.90755 

.49071 

.961351 

23 

24 

.44661 

.80704 

.46123 

.85608 

.47601 

.90845 

.49096 

.964-18 

24 

25 

.44685 

.80783 

.46147 

.85692 

.47626 

.90935 

.49121 

.96544 

25 

26 

.44709 

.80862 

.46172 

.  85777 

.47651 

.91026 

.49146 

.96641 

26 

27 

.44734 

.80942 

.46196 

.85861 

.47676 

.91116 

.49171 

.967'38 

27 

28 

.44758 

.81021 

.46221 

.85946 

.47701 

.91207 

.49196 

.96835 

28 

29 

.44782 

.81101 

.46246 

.86031 

.47725 

.91297 

.49221 

.96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246 

.97029 

30 

31 

.44831 

.81260 

.46295 

.86201 

.47775 

.91479 

.49271 

.97127 

31 

32 

.44855 

.81340 

.46319 

.86286 

.47800 

.91570 

.49296 

.97224 

32 

33 

.44879 

.81419 

.46344 

.86371 

.47825 

.91681 

.49321 

.97322 

88 

34 

.44903 

.81499 

.46368 

.86457 

.47849 

.91752 

.49346 

.97420 

34 

35 

.44928 

.81579 

.46393 

.865-12 

.47874 

.91844 

.49372 

.97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899 

.91935 

.49397 

.97615 

36 

37 

.44976 

.81740 

.46442 

.86713 

.47924 

.92027 

.49422 

.97713 

37 

38 

.45001 

.81820 

.46466 

.86799 

.47949 

.92118 

.49447 

.97811 

38 

39 

.450-25 

.81900 

.46491 

.86885 

.47974 

.92210 

.49472 

.97910 

39 

40 

.45049 

.81981 

.46516 

.86990 

.47998 

.92302 

.49497 

.98008 

40 

41 

.45073 

.82061 

.46540 

.87056 

.48023 

.92394 

.49522 

.98107 

41 

42 

.45098 

.82142 

.46565 

.87142 

.48048 

.92486 

.49547 

.98205 

42 

43 

.45122 

.82222 

.46589 

.87229 

.48073 

.92578 

.49572 

.98304 

43 

44 

.45146 

.82303 

.46614 

.87315 

.48098 

.92670 

.49597 

.98403 

44 

45 

.45171 

.82384 

.46639 

.87401 

.48123 

.92762 

.49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488 

.48148 

.92855 

.49648 

.98601 

46 

47 

.45219 

.82546 

.46688 

.87574 

.48172 

.92947 

.49673 

.98700 

47 

48 

.45244 

.82627 

.46712 

.87661 

.48197 

.93040 

.49698 

.98799 

48 

49 

.45268 

.82709 

.46737 

.87748 

.48222 

.93133 

.49723 

.98899 

49 

50 

.45292 

.82790 

.46762 

.87834 

.48247 

.93226 

.49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

.49773 

.99098 

51 

52 

.45341 

.82953 

.46811 

.88008  ' 

.48207 

.93412 

.49799 

.99198 

52 

53 

.45365 

.83034 

.46886 

.88095 

.4R-J22 

.93505 

.49824 

.99298 

53 

54 

.45390 

.83116 

.46860 

.88183 

.48347 

.93598 

.49849 

.99398 

54 

55 

.45414 

.83198 

.46885 

.88270 

.4*372 

.93692 

.49874 

.99498 

55 

56 

.45439 

.83280 

.46909 

.88357 

.48396 

.93785 

.49899 

.99598 

56 

57 

.45463 

.83362 

.46934 

.88445 

.48421 

.93879 

.49924 

.99698 

57 

58 

.45487 

.83444 

.46959 

.88532 

.484-16 

.9397-3 

.49950 

.99799 

58 

59 

.45512 

.83538 

.469S3 

.88(530 

.48471 

.94066 

!  49875 

.99899 

59 

60 

.45536 

.83608  i 

.47008 

.88708 

[48406 

.941  CO 

.50000 

i.  00000 

60 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


347 


1 

€ 

0° 

c 

1° 

|   e 

2° 

6 

3° 

Vers. 

Exsec. 

Vers. 

j  Exsec. 

Yers. 

Exsec. 

Vers. 

Kxsec. 

0 

1  .50000 

1  i.ooooo 

:  .51519 

1  1.06267 

.53053 

1.13005 

.54601 

1.20269 

0 

1 

.50025 

1.00101 

.51544 

1.06375 

.53079 

1.13122 

.54627 

1.20395 

1 

2 

.50050 

1.00202 

.51570 

1.06483 

.53104 

1.13239 

.54653 

1.20521 

2 

8 

.50076 

1.00303 

.51595 

1.06592 

.53130 

1.13356 

.54679 

1.20647 

3 

4 

.50101 

1.00404 

.51621 

1.06701 

.53156 

1.13473 

.54705 

1.20773 

4 

5 

.50126 

1.00505 

.51646 

1.06809 

.53181 

1.13590 

.54731 

1.20900 

5 

G 

.50151 

1.00607 

.51672 

1.06918 

.53207 

1.13707 

.54757 

1.21026 

6 

7 

.50176 

1.00708 

.51697 

1.07027 

.53233 

1.13825 

.54782 

1.21153 

8 

.50202 

1.00810 

.51723 

1.07137 

.53258 

1.13942 

.54808 

1.21280 

8 

9 

.50227 

1.00912 

.51748 

1.07246 

.53284 

1.14060 

.54834 

1.21407 

9 

10 

.50252 

1.01014 

.51774 

1.07356 

.53310 

1.14178 

.54860 

1.21535 

10 

11 

.50277 

1.01116 

.51799 

1.07465 

.53336 

1.14296 

.54886 

1.21662 

11 

1-3 

.50303 

1.01218 

.51825 

.07575 

.53361 

1.14414 

.54912 

1.21790 

12 

13 

.50328 

1.01320 

.51850 

.07685 

.53387 

1.14533 

.54938 

1.21918 

13 

14 

.50353 

1.01422 

.51876 

.07795 

.53413 

1.14651 

.54964 

1.22045 

14 

15 

.50378 

1.01525 

.51901 

.07905 

.53439 

.14770 

.54990 

1.22174 

15 

1(5 

.50404 

1.01628 

.51927 

.08015 

.53464 

.14889 

.55016 

1.22302 

16 

17 

.50429 

1.01730 

.51952 

.08126 

.53490 

.15008 

.55042 

1.22430 

17 

18 

.50454 

1.01833 

.51978 

.08236 

.53516 

.15127 

.55068 

1.22559 

18 

19 

.50479 

1.01936 

.52003 

.08347 

.53542 

.15246 

.55094 

1.22688 

19 

20 

.50505 

1.02039 

.52029 

.08458 

.53567 

.15366 

.55120 

1.22817 

20 

21 

.50530 

1.02143 

.52054 

.^8569 

.53593 

.15485 

.55146 

1.22946 

21 

2° 

.50555 

1  02246 

.52080 

.08680 

.53619 

.15605 

.55172 

1.23075 

22 

23 

.50581 

1.02349 

.52105 

.08791 

.53645 

.15725 

.55198 

1.23205 

23 

24 

.50606 

1.02453 

.52131 

.08903 

.53670 

.15845 

.55224 

1.23334 

24 

25 

.50631 

1.02557 

.52156 

.09014 

.53696 

.15965 

.55250 

1.23464 

25 

26 

.50656 

1.02661 

.52182 

.09126 

.53722 

.16085 

.55276 

1.23594 

26 

27 

.50682 

1.02765 

.52207 

.09238 

.53748 

.16206 

!  .55302 

1.23724 

°7 

28 

.50707 

1.02869 

.52233 

.09350 

.53774 

.16326 

j  .55328 

1.23855 

28 

29 

.50732 

1.02973 

.52259 

.09462 

.53799 

.16447 

.55354 

1.23985 

29 

30 

.50758 

1.03077 

.52284 

.09574 

.53825 

.16568 

.55380 

1.24116 

30 

31 

.50783 

1.03182 

.52310 

.09686 

.53851 

.16689 

.55406 

1.24247 

31 

32 

.50808 

1.03286 

.52335 

.09799 

.53877 

.16810 

.55432 

1.24378 

32 

33 

.50834 

1.03391 

.52361 

.09911 

.53903 

.16932 

.55458 

1.24509 

&3 

84 

.50859 

1.03496 

.52386 

.10024 

.53928 

.17053 

.55484 

1.24640 

34 

36 

.50884 

1.03601 

.52412 

.10137 

.53954 

.17175 

.55510 

1.24772 

35 

36 

.50910 

1.03706 

.52438 

.10250 

.53980 

.17297 

.55536 

1.24903 

36 

V9 

.50935 

1.03811 

.52463 

.10363 

.54006 

.17419 

.55563 

1.25035 

37 

38 

.50960 

1.03916 

.52489 

.10477 

.54032 

.17541 

.55589 

1  25167 

38 

39 

.50986 

1.04022 

.52514 

.10590 

.54058 

.17663 

.55615 

1.25300 

39 

40 

.51011 

1.04128 

.52540 

.10704 

.54083 

.17786 

.55641 

1.25432 

40 

41 

.51036 

1.  04233 

.52566 

.10817 

.54109 

.17909 

.55667 

1.25565 

41 

42 

.51062 

1.04339 

.52591 

.10931 

.54135 

.18031 

.55693 

1.25697 

42 

48 

.51087 

1.04445 

.52617 

.11045 

.54161 

.18154 

.55719 

1.25830 

43 

44 

.51113 

1.04551 

.£2642 

.11159 

.54187 

.18277 

.55745 

1.25963 

44 

45 

.51138 

1.04658 

.52668 

.11274 

.54213 

.18401 

.55771 

1.26097 

45 

40 

.51163 

1.04764 

.52694 

.11388 

.54238 

.18524 

.55797 

1.26230 

46 

17 

.51189 

1.04870 

.52719 

.11503 

.54264 

.18648 

.55823 

1.26364 

47 

48 

.51214 

1.04977 

.52745 

.11617 

.54290 

.18772 

.55849 

1.26498 

48 

49 

.51239 

1.05084 

.52771 

.11732 

.54316 

.18895 

.55876 

1.26632 

49 

50 

.51265 

1.05191 

.52796 

.11847 

.54342 

.19019 

.55902 

1.26766 

50 

51 

.54290 

1.05298 

.52822 

.11963 

.54368 

.19144 

.55928 

1.26900 

51 

52 

.51316 

1.05405 

.528-18 

.12078 

.54394 

.19268 

.55954 

1.27035 

52 

53 

.51341 

1.05512 

.52873 

.12193 

.54420 

.19393 

.55980 

1.27169 

53 

54 

.51366 

1.05619 

.52899 

.12309 

.54446 

.19517 

.56006 

1.27304 

54 

55 

.51392 

1.05727 

.52924 

.12425 

.54471 

.19642 

.56032 

1.27439 

55 

50 

.51417 

1.05835 

.52950 

.12.540 

.54497 

.197'67 

.56058 

1.27574 

56 

57 

.51443 

1.05942 

.52976 

.12657 

.54523 

19892 

.56084 

1.27710 

57 

58 

.51468 

1.06050 

.53001 

.12773 

.54549 

20018 

.56111 

1.27845 

58 

51) 

.51494 

1.06158 

.53027 

.12889 

.54575 

1.20143 

.56137 

1.27981 

59 

60 

.51519 

1.06267 

.53053 

.13005  1 

.54601 

1.20269 

.56163 

1.28117 

60 

TABLE  xm.— VERSINES  AND  EXSEC  ANTS. 


6 

40 

6 

5° 

• 

6-     ! 

6 

7° 

Vers. 

i 
Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.56163 

1.28117  ; 

.57738 

1.36620 

.59326 

1.45859 

.60927 

.55930 

0 

1 

.56189 

1.28253  i 

.57765 

1.36768 

.59:353 

1.46020 

.60954 

.56106 

1 

2 

.56215 

1.28390  1 

.57791 

1.36916 

.59379 

1.46181 

.60980 

.56282 

2 

3 

.56241 

1.28526 

.57817 

1.37064 

.59406 

.46342 

.61007 

.56458 

3 

4 

.56267 

1.2S003 

.57844 

1.37212 

.59433 

.46504 

.61034 

.566:34 

4 

5 

.56294 

1.28800 

.57870 

1.37361 

.59459 

.46665 

.61061 

.56811 

5 

6 

.56320 

1.28937 

.57896 

1.37509 

.59486 

.46827 

.61088 

.56988 

6 

7 

.56346 

1.29074  i 

.57923 

1.37658  : 

.59512 

.46989 

.61114 

.57165 

7 

8 

.56372 

1.29211  ! 

.57949 

1.37808 

.59539 

.47152 

.61141 

.57342 

8 

9 

.56398 

1.29349 

.57976 

1.37957 

.59566 

.47314 

.61168 

.57520 

9 

10 

.56425 

1.29487 

.58002 

1.38107' 

.59592 

.47477 

.61195 

.57698 

10 

11 

.56451 

1.29625 

.58028 

1.  138256 

.59619 

.47640 

.61222 

.57876 

11 

1-2 

.56477 

1-29763 

.58055 

1.38406 

.59645 

.47804 

.61248 

.58054 

12 

13 

.56503 

1.29901  1 

.58081 

1.38556 

.5967'2 

.47967 

.61275 

.58233 

13 

14 

.56529 

1.30040 

.58108 

1.38707 

.59699 

.48131 

.61302 

.58412 

14 

15 

.56555 

1.30179 

.58134 

1.38857 

.59725 

.48295 

.61329 

.58591 

15 

16 

.56582 

1.30318  ! 

.58160 

1.39008 

.59752 

.48459 

.61356 

.58771 

16 

17 

.56608 

1.30457 

.58187 

1.39159 

.59779 

.48624 

.61383 

.58950 

17 

18 

.56634 

1.30596 

.58213 

1.39311 

.59805 

.48789 

.61409 

.59130 

18 

19 

.56660 

1.30735 

.58240 

1.39462 

.59832 

.48954 

.61436 

.59311 

19 

20 

.56687 

1.30875 

.58266 

1.39614 

.59859 

.49119 

.61463 

.59491 

20 

21 

.56713 

1.31015 

.58293 

1.39766 

.59885 

.49284 

.61490 

.59672 

21 

22 

.56739 

1.31155  | 

.58319 

1.39918 

.59912 

.49450 

.61517 

.59853 

22 

23 

.56765 

1.31295  i 

.58345 

1.40070 

.599:38 

.49616  j 

.61544 

.60035 

23 

24 

.5679! 

1.31436 

.58372 

1.40222 

.59965 

.49782  | 

.61570 

.60217 

24 

25 

.56818 

1.31576 

.58398 

1.40375 

.59992 

.49948 

.61597 

.60399 

25 

20 

.56844 

1.31717 

.58425 

1.40528 

.60018 

.50115 

.61624 

.60581 

26 

27 

.56870 

1.31858 

.58451 

1.40681 

.60045 

.50282 

.61651 

.60763 

27 

28 

.56896 

1.31999 

.58478 

1.40835  j 

.60072 

.50449 

.61678 

.60946 

28 

2!) 

.56923 

1.32140 

.58504 

1.40988  1 

.60098 

.50617 

.61705 

.61129 

29 

80 

.56949 

1.32282 

.58531 

1.41142 

.60125 

.50784 

.61732 

.61313 

30 

31 

.56975 

1.32424 

.58557 

1.41296 

.60152 

.50952  ' 

.61759 

.61496 

31 

32 

.57001 

1.32566 

.58584 

1.41450 

.60178 

.51120 

.61785 

.61680 

32 

88 

.57028 

1.32708 

.58610 

1.41605 

.60205 

.51289 

.61812 

.61864 

33 

84 

.57054 

1.32850 

.58637 

1.41760 

.60232 

.51457 

.61839 

.62049 

34 

86 

.57080 

1.32993 

.58663 

1.41914 

.60259 

.51626 

.61866 

.62234 

35 

86 

.57106 

1.331.35 

.58690 

1.42070 

.60285 

.51795 

.61893 

.62419 

36 

37 

.57133 

1.33278 

.58716. 

1.42225 

.60312 

.51965 

.61920 

.62604 

37 

3S 

.57159 

1.33422 

.58743 

1.42380 

.60339 

.52134 

.61947 

.62790 

38 

39 

.57185 

1.33565 

.58769 

1.42536 

.60365 

.52304 

.61974 

.62976 

39 

40 

.57212 

1.33708 

.58796 

1.42692 

.60392 

.52474 

.62001 

.63162 

40 

41 

.57238 

1.33852 

.58822 

1.42848 

.60419 

.52645 

.62027 

.63348 

41 

42 

.57264 

1.33996 

.58849 

1.43005 

.60445 

.52815  ; 

.62054 

.63535 

42 

43 

.57291 

1.34140 

.58875 

1.43162 

.60472 

.52986 

.62081 

.63722 

43 

44 

.57317 

1.34284 

.58902 

1.43318 

.60499 

.53157 

.62108 

.63909 

44 

45 

.57343 

1.34429 

.58928 

1.43476 

.60526 

.53329 

.62135 

.64097 

45 

46 

.57369 

1.34573 

.58955 

1.43633 

.60552 

.53500 

.62162 

.64285 

46 

47 

.57396 

1.34718 

.58981 

1.43790 

.60579 

.53672 

.62189 

.64473 

47 

48 

.57422 

1.34863 

.59008 

1.43948 

.60606 

.538-15 

.62216 

.64662 

48 

59 

.57448 

1.35009 

.59034 

1.44106 

.60633 

.54017 

.62243 

.64851 

49 

50 

.57475 

1.35154 

.59061 

1.44264 

.60659 

.54190 

.62270 

.65040 

50 

51 

.57501 

1.35300 

.59087 

1.44423 

.60686 

.54363 

.62297 

.65229 

51 

52 

.57527 

1.35446 

.59114 

1.44582 

.60713 

.54536 

.62324 

.65419 

52 

53 

.57554 

1.35592 

.59140 

1.44741 

.60740 

.54709 

.62351. 

.65609 

53 

54 

.57580 

1.35738 

.59167 

1.44900 

.60766 

.54883 

.62378 

.65799 

54 

55 

.67^06 

1.35885 

.59194 

1.45059 

.60793 

.5505? 

.62405 

.65989 

55 

50 

.57833 

1.36031 

.59220 

1.45219 

.60820 

.55231 

.62431 

.66180 

50 

57 

.57659 

1.36178 

.59247 

1.45378 

.60847 

.55405 

.62458 

.66371 

57 

5H 

.57685 

1.36325 

.59273 

1.45539 

.60873 

.55580 

.62485 

.66568 

58 

59 

.57712 

1.3(1473 

.59300 

1.45699 

.60900 

1.55755 

.62512 

1.60755 

59 

GO 

.57738 

1.36620 

.59326 

1.45859 

.60927 

1.55930 

.62539 

1.66947 

60 

TABLE  XIII.— VERSINES  AND  EXSEOANTS. 


349 


/ 

68'          69° 

70° 

71° 

' 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.62539  !  1.66947 

.64163 

1.79043 

.65798 

1.92380 

.67443 

2.07155 

0 

1 

.62566   1.67139 

.64190 

1.79254 

.65825 

1.92614 

.67471 

2.07415 

1 

2 

.62593 

1.67332 

'.64218 

1.79466 

.65853 

1.92849 

.67498 

2.07675 

2 

3 

.62620 

1.67525 

.64245 

1.79679 

.65880 

1.93083 

.67526  2.07936 

4 

.62647  1.67718 

.64272 

1.79891 

.65907 

1.93318 

.67553 

2.08197 

4 

5 

.62674   1.67911 

.64299 

1.80104 

.65935 

1.93554 

.67581 

2.08459 

5 

6 

.62701   1.68105 

.64326 

1.80318 

.65962 

1.93790 

.67608 

2.08721 

6 

7 

.62728   1.68299 

.64353 

1.80531 

.65989 

1.94026 

.67636 

2.08983 

7 

8 

.62755   1.68494 

.64381 

1.80746 

•  .66017 

1.94263 

.67663 

2.09246 

8 

9 

.62782  1.68689 

.64408 

1.80960 

.66044 

1.94500 

.67691 

2.09510 

9 

10 

.62809 

1.68884 

.64435 

1.81175 

j  .66071 

1.94737 

.67718 

2.09774 

10 

11 

.62836 

1  69079 

.64462 

1.81390 

1  .66099 

1.94975 

.67746 

2.10038 

11 

12 

.62863   1.69275  1  .64489 

1.81605 

.66126 

1.95213 

.67773 

2.10303 

12 

13 

.62890   1.69471   .64517 

1.81821 

.66154 

1.95452 

.67801 

2.10568 

13 

14 

.62917 

1.69667 

.64544 

1.82037 

.66181 

1.95691 

.67829 

2.10834 

14 

15 

.62944 

1.69864 

.64571 

1.82254 

.66208 

1.95931 

.67856 

2.11101 

15 

16 

.629?!   1.70061 

.64598 

1.82471 

.6G236 

1.96171 

.67884 

2.11367 

16 

17 

.62998  1.70258 

.64625 

1.8.2688 

.66263 

1.96411 

.67911 

2.11635 

17 

18 

.63025   1.70455 

.64653 

1.82906 

.66290 

1.96652 

.67939 

2.11903 

18 

19 

.63052   1.70653 

.64680 

1.83124 

;  .66318 

1.96893 

.(57966 

2.12171 

19 

20 

.63079  1.70851 

.64707 

1.83342 

|  .66345 

1.97135 

.67994 

2.12440 

20 

21 

.63106   1.71050 

.64734 

1.83561 

.66373 

1.97377 

.68021 

2.12709 

21 

22 

.63133   1.71249 

.64761 

1.83780 

.66400 

1.97619 

.68049 

2.12979 

22 

'23 

.63161   1.71448 

.64789 

1.83999 

.66427 

1.97862 

.68077 

2.13249 

23 

24 

.63188 

1.71647 

.64816   1.84219  l|  .66455 

1.98106 

.68104 

2.13520 

24 

25 

.63215 

1.71847 

.64843 

1.84439   .66482 

1.98349 

.68132 

2.13791 

25 

26 

.63242 

1.72047 

.64870 

1.84659 

.66510 

1.9S594 

.68159 

2.14063 

26 

27 

.63269 

1.7'2247 

.64898  1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28 

.63290 

1.72448 

.64925  !  1.85102 

.66564 

1.99083 

.68214 

2.14608 

28 

29 

.63323 

1.72649 

.64952   1.85323 

.66592 

1.99329 

.68242 

2.14881 

29 

30 

.63350 

1.72850 

.64979  1.85545 

.66619 

1.99574 

.68270 

2.15155 

30 

31 

.63377 

1.73052 

.65007 

1.85767 

.66647 

1.99821 

.68297 

2.15429 

31 

32 

.63404 

1.73254 

.65034 

1.85990 

.66G74 

2.000C7 

.68325 

2.15704 

32 

33 

.63431 

1.73456 

.65061   1.86213 

.667'02 

2.00315 

.68352 

2.15979 

33 

34 

.63458 

1.73659  j 

.65088   1.86437 

.66729 

2.00562 

.68380 

2.16255 

34 

35 

.63485 

1.73862  i 

.65116   l.BtiGlil 

.66756 

2.00810 

.68408 

2.16531 

35 

36 

.63512 

1.74065 

.65143   1.86885 

.66784 

2.01059 

.68435 

2.16808 

36 

37 

.63539 

1.74269  I 

.65170  1.87109 

.66811 

2.01308  | 

.68463 

2.17085 

37 

38 

.63566 

1.74473 

.65197 

1.87334 

.66839 

2.01557 

.68490 

2.17363 

38 

39 

.63594 

1.74677 

.65225 

1.87560 

.66866 

2.01807 

.68518 

2.17641 

39 

40 

.63621 

1.74881 

.65252 

1.87785 

.66894 

2.02057 

.68546 

2.17920 

40 

41 

.63648 

1.75086 

.65279 

1.88011 

.66921 

2.02308 

.68573 

2.18199 

41 

42 

.63675 

1.75292 

.65:306   1.88238 

.66949 

2.02559 

.68601 

2.18479 

42 

43 

.63702 

1.75497 

.65334 

1.88465 

.6697'6 

2.02810 

.68628 

2.18759 

43 

44 

.63729 

1.75703 

.65361 

1.88692 

.67003 

2.03062 

.68656 

2.19040 

44 

45 

.63756 

1.75909 

.65388 

1.88920 

.67'031 

2.03315 

.68684 

2.19322 

45 

46 

.63783 

1.76116 

.65416  1.89148 

.67058 

2.03568 

.68711 

2.19604 

46 

47 

.63810 

1.76323 

.65443 

1.89376 

.67086 

2.03821 

.68739 

2.19886 

47 

48 

.63838   1.76530  j 

.65-170 

1.89605 

.67113 

2.04075 

.68767 

2.20169 

48 

49 

.6-3865 

1.76737 

.65497 

1.89834 

.67141 

2.04329 

.68794 

2.20453 

49 

50 

.63892 

1.76945 

.65525 

1.90063 

.67168 

2.04584 

.68822 

2.20737 

50 

51 

.63919 

1.77154 

.65552 

1.90293 

.67196 

2.04839 

.68849 

2.21021 

51 

52 

.63946 

1.77362 

.65579   1.90524 

.67223 

2.05094 

.68877 

2.21306 

52 

53 

.63973 

1  .  77571 

.65607 

1.90754 

.67251 

2.05350 

.68905 

2.21592 

53 

54 

.61000 

1  77780 

.65634 

1.90986 

.67278 

2.05607 

.68932 

2.21878 

54 

55 

.64027 

1.77990 

.65661 

1.91217 

.67306 

2.05864 

.68960 

2.2-3165  55 

56 

.64055  I  1.78200 

.65689 

1.91449 

.67333 

2.06121 

.68988 

2.22452  156 

57 

640S2   1.78410 

.65716 

1.91681 

.673(51   2.06379 

.69015 

2.22740  57 

58 

.64109 

1.7H621 

.65743 

1.91914 

.67388 

2.06637 

.69043 

2.23028 

58 

59 

.64136 

1.78833 

.65771 

1.92147 

.67416 

2.06896  ; 

.69071 

2.23317 

59 

60  1  .64103 

1.7-9043 

.65798 

1.92380 

.67443 

2.07155  II  .69098 

2.23607 

60 

350 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


' 

72° 

73° 

74° 

75° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.69098 

2.23607 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

0 

1 

.69126 

2.23897 

.70791 

2.42356 

.72464 

2.63164 

.74146 

2.86790 

1 

8 

.09154 

2.24187 

.70818 

2.42683 

.72492 

2  63533 

.74174 

2.87211 

2 

8 

.69181 

2.24478 

.70846 

2.43010 

.72520 

2.63903 

.74202 

2.87633 

8 

4 

.69209 

2.24770 

.70874 

2.43337 

.72518 

2.64274 

.74231 

2.88056 

4 

5 

.69237 

2.25062 

.70902 

2.43666 

.72576 

2.64645 

.  71259 

2.88479 

5 

6 

69264 

2.25355 

.70930 

2.43995 

.72604 

2.65018 

.74287 

2.88904 

6 

7 

.69292 

2.25648 

.70958 

2.44324 

.72632 

2.65391 

.74315 

2.89330 

7 

8 

.69320 

2.25942 

.70985 

2.44655 

.72660 

2.657'65 

.74848 

2.89756 

8 

9 

.69347 

2  26237 

.71013 

2.44986 

.72688 

2.66140 

.74371 

2.90184 

9 

10 

.69375 

2.26531 

.71041 

2.45317 

.72716 

2.66515  : 

.74399 

2.90613 

10 

11 

.69403 

2.26827 

.71069 

2.45650 

.72744 

2.66892 

.74427 

2.91042 

11 

12 

.69430 

2.27123 

.71097 

2.45983 

.72772 

2.67269 

.74455 

2.91473 

12 

18 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.67647 

.74484 

2.91904 

13 

1-1 

.69486 

2.27717 

.71153 

2.46651 

.72828 

2.68025  ! 

.74512 

2.92337 

14 

15 

.69514 

2.28015 

.71180 

2.46986 

.72856 

2.68405 

.74540 

2.92770 

15 

16 

.69541 

2.28313 

.71208 

2.47321 

.72884 

2.68785 

.74568 

2.93204 

16 

17 

.69569 

2.28612 

.71236 

2.47658 

.72912 

2.69167  i 

.74596 

2.9364C 

17 

18 

.69597 

2.28912 

.71264 

2.47995 

.72940 

2.  (.9549 

.74624 

2.94076 

18 

19 

.69624 

2.29212 

.71292 

2.48333 

.72968 

2.69931 

.74652 

2.94514 

19 

20 

.69652 

2.29512 

.71320 

2.48671 

.72996 

2.70815 

.74680 

2.94952 

20 

21 

.69680 

2.29814 

.71348 

2.49010 

.73024 

2.70700 

.74709 

2.95392 

21 

22 

.69708 

2.30115 

.71375 

2.49350 

.73052 

2.71085 

.74737 

2.95832 

22 

23 

.69735 

2.30418 

.71403 

2.49691 

.73080 

2.71471 

.74765 

2.96274 

23 

24 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858  I 

.747-93 

2.96716 

24 

25 

.69791 

2.31024 

.71459 

2.50374 

.73136 

2.72245  i 

.74821 

2.97160 

25 

26 

.69818 

2.31328 

.71487 

2.50716 

.73164 

2.72635  1 

.74849 

2.97604 

26 

27 

.69846 

2.31633 

.71515 

2.51060 

.73192 

2.73024 

.74878 

2.98050 

27 

28 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.73414  : 

.74906 

2.98497 

28 

29 

.69902 

2  32244 

.71571 

2.51748 

.73248 

2.73806 

.74934 

2.98944 

29 

30 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

30 

31 

.69957 

2.32858 

.71626 

2.52440 

,73304 

2.74591 

.74990 

2.998-13 

81 

32 

.69985 

2.33166 

.71654 

2.52787 

.73332 

2.74984 

.75018 

3.00293 

32 

88 

.70013 

2.33474 

.71682 

2.53134 

.73360 

2.75379 

.75047 

3.00745 

88 

3-1 

.70040 

2.33783 

.71710 

2.5:3482 

.73388 

2.75775 

.75075 

3.01198 

34 

35 

.70068 

2.34092 

.71738 

2.53831 

.73416 

2.76171 

.75103 

3.01652 

85 

36 

.70096 

2.34403 

.71766 

2.54181 

.73444 

2.76568 

.75131 

3.02107 

86 

37 

.70124 

2.34713 

.71794 

2.54531 

.73472 

2.76966 

.75159 

3.02563 

87 

38 

.70151 

2.35025 

.71822 

2.54883 

.73500 

2.77365 

!  78187 

3.03020 

38 

39 

.70179 

2.35336 

.71850 

2.55235 

.73529 

2.77765 

.75216 

3.03479 

3!) 

40 

.70207 

2.35649 

.71877 

2.55587 

.73557 

2.78166 

.75244 

3.03938 

40 

41 

.70235 

2.35962 

.71905 

2.55940 

.73585 

2.78568 

.75272 

3.04398 

41 

42 

.70263 

2.36276 

.71933 

2.56294 

.73613 

2.78970 

.75300 

3.04860 

42 

43 

.70290 

2.36590 

.71961 

2.56649 

.73641 

2.79374 

.75328 

3.05322    43 

44 

.70318 

2.36905 

.71989 

2.57005 

.73669 

2.79778 

.75356 

3.05786 

44 

45 

.70346 

2.37221 

.72017 

2.57361 

.73697 

2.80183 

.75385 

3.06251 

45 

46 

.70374 

2.37537 

.72045 

2.57718 

.73725 

2.80589 

.7'5413 

3.06717 

4<i 

47 

.70401 

2.37854 

.72073 

2.5807'6 

.73753 

2.80996 

.75441 

3.07184 

47 

48 

.70429 

2.38171 

.72101 

2.58434 

.73781 

2.81404 

.75469 

3.07652 

48 

49 

.70457 

2.38489 

.72129 

2.58794 

.73809 

2.81813 

.75497 

3.08121 

48 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223 

.75526 

3.08591 

50 

51 

.70513 

2.39128 

.72185 

2.59514 

.73865 

2.826&3 

.75554 

3.09063 

51 

52 

.70540 

2.39448 

.72213 

2.59876 

.73893 

2.83045 

.75582 

3.09535    52 

53 

.70568 

2.39768 

.72241 

2.60238 

.73921 

2.83457 

.75610 

3.10009  |53 

54 

.70596 

2.40089 

.72269  !  2.60601 

.73950 

2.83871 

.75639 

3.10484    54 

55 

.70624 

2.40411 

.72296     2.60965 

.73978 

2.84285 

.75667 

3.10960    55 

56 

.70652 

2.40734 

.72324 

2.61330 

.71006 

2.84700 

.75695 

3.11437  !56 

57 

.70679 

2.41057 

72352 

2.61695 

.74034 

2.85116 

.75723 

3.11915  J57 

58     .7'1707 

2.41381 

.72:580  '  2.62061 

.740(52 

2.85533  i     .75751      3.12394 

58 

.r!>    .nra 

2.41705 

.72408      2.62128 

.74090 

2  S5951       .75780 

8.12875 

59 

CO    .70763 

2.42030 

.72436  !  2.62796 

.74118     2.86370       .75808 

3.13357  i60 

TABLE  XIII.—  VERSINES  AND   EXSECANTS. 


351 


' 

76°          77°          78°          79° 

• 

Vers. 

Exsec. 

Vers.  Exsec. 
i 

Vers. 

Exsec. 

Vers. 

Exsec. 

G!  .75808 

3.13357 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

0 

1 

.75836 

3.13839 

.77533 

3.45102 

.79237 

3.81633 

.80948 

4.24870 

1 

2 

.75864 

3.14323 

.77562 

3.45664 

.79266 

3.82294 

.80976 

4.25658 

2 

3 

.75892 

3.14809 

.  77590 

3.46228 

.79294 

3.82956 

.81005 

4.26448 

3 

4 

.75921 

3.15295 

.77618 

3.46793 

.79323 

3.83621 

.81033 

4.27241 

4 

5 

.75949 

3.15782 

.77647 

3.47360 

.79351 

3.84288 

.81062 

4.28036 

5 

G 

.75977 

3.16271 

.  77675 

3.47'928 

.79380 

3.84956 

.81090 

4.28833 

6 

7 

.76005 

3.16761 

.77703 

3.48498 

.79408 

3.85627 

.81119 

4.29634 

7 

8 

.76034 

3.17252 

.77732 

3.49069 

.79437 

3.86299 

.81148 

4.30436 

8 

9 

.76062 

3.17744 

.77760 

3.49642 

.79465 

3.86973 

.81176 

4.31241 

9 

10 

.76090 

3.18238 

.77788 

3.50216 

.79493 

3.87649 

.81205 

4.32049 

10 

11 

.76118 

3.1S733 

.77817 

3.50791 

.79522 

3.88327 

.81233 

4.32859 

11 

12 

.76147 

3.19228 

.77845 

3.51368 

.79550 

3.89007 

.81262 

4.33671 

12 

13 

.76175 

3.19725 

.  7787'4 

3.51947 

.79579 

3.89689 

.81290 

4.34486 

13 

14 

.76203 

3.20224 

.77902 

3.52527 

.7'9607 

3.90373 

.81319 

4.35304 

14 

15 

.76231 

3.207.23 

.77930 

3.53109 

.79636 

3.91058 

.81348 

4.36124 

15 

16 

.76260 

3.21224 

.77959 

3.53692 

.79664 

3.917'46 

.81376 

4.36947 

16 

17 

.76288 

3.21726 

.77987 

3.54277 

.79693 

3.92436 

.81405 

4.37772 

17 

18 

.76316 

3.22229 

.78015 

3.54863 

.79721 

3.93128 

.81433 

4.38600 

18 

19 

.76344 

3.22734 

.7'8044 

3.55451 

.79750 

3.93821 

.81462 

4.394*) 

19 

20 

.76373 

3.23239 

.78072 

3.56041 

.79778 

3.94517 

.81491 

4.40263 

20 

21 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099 

21 

.22 

.76429 

3.24255 

.7-8129 

3.57224 

.79335 

3.95914 

.81548 

4.41937 

22 

23 

.76458 

3.24764 

.78157 

3.57819 

.79864 

3.96616 

.81576 

4.42778 

23 

24 

.76486 

3.25275 

.78186 

3.58414 

.79892 

3.97320 

.81605 

4.43622 

24 

25 

.76514 

3.25787 

.78214 

3.59012 

.79921 

3.98025 

.81633 

4.44468 

25 

20 

.,76542 

3.26300 

.78242 

3.59611 

.79949 

3.98733 

.81662 

4.45317 

26 

27 

.  76571 

3.26814 

.78271 

3.60211 

.79978 

3.99443 

.81691 

4.46169 

27 

28 

.76599 

3.27330 

.78299 

3.60813 

.80006 

4.00155 

.81719 

4.47023 

28 

29 

.76627 

3.27847 

.78328 

3.61417 

.80035 

4.00869 

.81748 

4.47881 

29 

30 

.76655 

3.28366 

.78356 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

80 

31 

.76684 

3.28885 

.78384 

3.62630 

.80092 

4.02303 

.81805 

4.49603 

31 

32 

.76712 

3.29406 

.78413 

3.63238 

.80120 

4.03021 

.81834 

4.50466 

32 

33 

.76740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

.81862 

4.51337 

33 

84 

.76769 

3.30452 

.78470 

3.64461 

.80177 

4.04471 

.81891 

4.52208 

34 

35 

.76797 

3.30977 

.78498 

3.65074 

.80206 

4.05197 

.81919 

4.53081 

35 

36 

.76825 

3.31503 

.78526 

3.65690 

.80234 

4.05926 

.81948 

4.53958 

36 

37 

.76854 

3.32031 

.7B555 

3.66307 

.80263 

4.06657 

.81977 

4.54837 

37 

38 

.76882 

3.32560 

.78583 

3.66925 

.80291 

4.07390 

.82005 

4.55720 

38 

39 

.76910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605 

39 

40 

.76938 

3.33622 

.78640 

3.68167 

.80348 

4.08863 

.82063 

4.57493 

40 

41 

.76967 

3.34154 

.78669 

3.68791 

.80377 

4.09602 

.82091 

4.58383 

41 

42 

.76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

.82120 

4.59277 

42 

43 

.77023 

3.35224 

.78725 

3.70044 

.80434 

4.11088 

.82148 

4.6017'4 

43 

44 

.77052 

3.35761 

.78754 

3.70673 

.80462 

4.11835 

.82177 

4.61073 

44 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61976 

45 

46 

.77108 

3.36839 

.7-8811 

3.71935 

.80520 

4.13334 

.82234 

4.62881 

46 

47 

.77137 

3.37'380 

.78839 

3.72569 

.80548 

4.14087 

.82263 

4.63790 

47 

48 

.77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292 

4.64701 

48 

49 

.77193 

3.38466 

.78896 

3.7-3843 

.80605 

4.15599 

.82320 

4.65616 

40 

50 

.77222 

3.39012 

.78924 

3.74482 

.80634 

4.16359 

.82349 

4.66533 

50 

51 

.77250 

3.39558 

.78953 

3.75123 

.80662 

4.17121 

.82377 

4.67454 

51 

52 

.77'278 

3.40106 

.78981 

3.7'5766 

.80691 

4.17886 

.82406 

4.68377 

52 

53 

.77307 

3.40656 

.79010 

3.7'6411 

.80719 

4.18652 

.82435 

4.GCS04  53 

54 

.77335 

3.41206 

.79038  3.77057 

.80748 

4.19421 

.82463 

4.70234  |54 

55 

.77363 

8.4175P 

.79067  i  3.77705 

.80776 

4.20193 

.82492 

4.71166  155 

56 

.7739S   S.  4231  2 

.79095 

3.78355 

.80805 

4.20966 

82521 

4.72102 

56 

57 

.77420 

3.42867 

.79123 

3.79007 

.soass 

4.21742 

82549 

4.73041 

57 

58 

.77448 

3.43424 

.79152 

3.79661 

.80862 

4.22521 

.82578 

4.73983  !  58 

59 

.77477 

3.43982 

.79180 

3.80316 

.80891 

4.23301 

.82607 

4.74929 

59 

60 

.77'505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

.82635  I  4.75877 

60 

352 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


i 

0° 

8 

i.   ! 

!  • 

2° 

8 

B 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.82635 

4.75877 

.84357 

5.39245  i 

.86083 

6.18530 

.87813 

7.20551 

0 

1 

.82664 

4.76829 

.84385 

5.40422 

.86112 

6.20020 

.87842 

7.22500 

1 

2 

.82692 

4.77784 

.84414 

5.41602 

.86140 

6.21517 

.87871 

7.24457 

2 

3 

.82721 

4.78742 

.84443 

5.42787 

.86169 

6.23019 

.87900 

7.26425 

3 

4 

.82750 

4.79703 

.84471 

5.43977 

.86198 

6.24529 

.87929 

7.28402 

4 

5 

.82778 

4.8066? 

.84500 

5.45171 

.86227 

6.26044 

.87957 

7.36388 

5 

6 

.82807 

4.81635 

.84529 

5.46369 

.86256 

6.27566 

.87986 

7.32384 

6 

7 

.82836 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.34390 

7 

8 

.82884 

4.83581 

.84586 

5.48779 

.86313 

6.30630 

.88044 

7.36405 

8 

9 

.82893 

4.84558 

.84615 

5.49991 

.86,542 

6.32171 

.88073 

7.38431 

9 

10 

.82922 

4.85539 

.84644 

5.51208 

.86371 

6.33719 

.88102 

7.40466 

10 

11 

.82950 

4.86524 

.84673 

5.52429 

.86400 

6.35274 

.88131 

7.42511 

11 

12 

.82979 

4.87511 

.84701 

5.53655 

.86428 

6.36835 

.88160 

7.44566 

12 

13 

.83003 

4.88502 

.84730 

5.54886 

.86457 

6.38403 

.88188 

7.46632 

13 

14 

.83036 

4.89497 

.84759 

5.56121 

.86486 

6.39978 

.88217 

".48707 

14 

15 

.83065 

4.90495 

.84788 

5.57361 

.86515 

6.41560 

.88246 

'.50793 

18 

10 

.83094 

4.91496 

.84816 

5.58606 

.86544 

6.43148 

.88275 

".52889 

10 

17 

.83122 

4.92501 

.84845 

5.59855 

.86573 

6.44743 

.88304 

'.54996 

17 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46346 

.88333 

".57113 

18 

19 

.83180 

4.94521 

.84903 

5.62369 

.86630 

6.47955 

.88362 

".59241 

19 

20 

.83208 

4.95536 

.84931 

5.63633 

.86659 

6.49571 

.88391 

'.61379 

20 

21 

.83237 

4.96555 

.84960 

5.64902 

.86688 

6.51194 

.88420 

".63528 

21 

22 

.83266 

4.97577 

.84989 

5.60176 

.86717 

6.52825 

.88448 

".65088 

22 

23 

.83294 

4.98603 

.85018 

5.67454 

.86746 

6.54462 

.88477 

7.67859 

23 

24 

.83323 

4.99633 

.85046 

5.68738 

.86774 

6.56107 

.88506 

7.70041 

24 

25 

.83352 

5.00666 

.85075 

5.70027 

.86803 

6.57759 

.88535 

".72234 

25 

20 

.83380 

5.01703 

.85104 

5.71321 

.86832 

6.59418 

.88564 

".74438 

26 

27 

.83409 

5.02743 

.85133 

5.72620 

.86861 

6.61085 

.88593 

'.76653 

27 

28 

.83438 

5.0378? 

.85162 

5.73924 

.86890 

6.62759 

.88622 

7.78880 

28 

2-9 

.83467 

5.04834 

.85190 

5.75233 

.86919 

6.64441 

.88651 

".81118 

29 

30 

.83495 

5.05886 

.85219 

5.76547 

.86947 

6.66130 

.88680 

'.83367 

30 

31 

.83524 

5.06941 

.85248 

5  .'77866 

.86976 

6.67826 

.88709 

".85628 

31 

32 

.83553 

5.08000 

.85277 

5.79191 

.87005 

6.69530 

.88737 

".87901 

32 

33 

.83581 

5.09062 

.85305 

5.80521 

.87034 

6.71242 

.88766 

'.90186 

33 

34 

.83610 

5.10129 

.85334 

5.81856 

.87063 

6.72962 

.88795 

'.92482 

34 

35 

.83639 

5.11199 

.85363 

5.&3196 

.87092 

6.74689 

.88824 

".94791 

35 

36 

.83667 

5.12273 

.85392 

5.84542 

.87120 

6.76424 

.88853 

".97111 

86 

37 

.83696 

5.13350 

.85420 

5.85893 

.87149 

6.78167 

.88882 

".99444 

37 

38 

83725 

5.14432 

.85449 

5.87250 

.87178 

6.79918 

.88911 

8.01788 

:$s 

39 

.83754 

5.15517 

.85478 

5.88612 

.87207 

6.81677 

.88940 

8.01140 

39 

40 

.83782 

5.16607 

.85507 

5.89979 

.87'236 

6.83443 

.88969 

8.06515 

40 

41 

.83811 

5.17700 

.85536 

5.91352 

.87265 

6.85218 

.88998 

8.08897 

41 

.83840 

5.18797 

.85564 

5.92731 

.87294 

6.87001 

.89027 

8.11292 

42 

43 

.83868 

5.19898 

.85593 

5.94115 

.87322 

6.88792 

.89055 

8.13099 

43 

44 

.83897. 

5.21004 

.85622 

5.95505 

.87351 

6.90592 

.89084 

8.16120 

44 

45 

.83926 

5.22113 

.85651 

5.96900 

.87380 

6.92400 

.89113 

8.18553 

45 

40 

.83954 

5.23226 

.85680 

5.98301 

.87409 

6.94216 

.89142 

8.20999 

40 

47 

.83983 

5.24343 

.85708 

5.99708 

.87438 

6.96040 

.89171 

8.23459 

47 

48 

.84012 

5.25464 

.8573? 

6.01120 

.87467 

6.97873 

.89200 

8.25931 

48 

4'J 

.84041 

5.20.V.H) 

.85700 

6.02538 

.87496 

6.99714 

.89229 

8.28417 

49 

50 

.84069 

5.27719 

.85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917 

50 

51 

.84098 

5.28853 

.85823 

6.05392 

.87553 

7.  03423 

.89287 

8.33430 

51 

52 

.84127 

5.29991 

.85852 

6.06828 

.87582 

7.05291 

.89316 

8  .  35957 

52 

53 

.84155 

5.31133 

.85881 

6.08269 

.87611 

7.07167 

.89345 

8.38497 

53 

54 

.84184 

5.32279 

.85910 

6.09717 

.87640 

7.09052 

.89374 

8.41052 

54 

55 

.84213 

5.33429 

.85939 

6.11171 

.87669 

7.10946 

.89403 

8.43620 

55 

50 

.84242 

5.34584 

.85967 

6.12630 

.87698 

7.12849 

.89431 

8.46203 

50 

57 

58 

.84270 
.84299 

5.35743 
5.36906 

.85996 
.86025 

6.14096 

6.15568 

.87726 
.87755 

7.14760 
7.16681 

.89460 
.89489 

8.48800 
8.51411 

57 
58 

.84328 

5.38073 

.86054 

6.17046 

.877*4 

7.18012 

.89518 

8.54037 

59 

60 

.84357 

5.39245 

.86083 

6.18530 

1  .87813 

7,20551 

1  .89547 

8.56677 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


353 


' 

84° 

85° 

86° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.89547 

8.56677 

.91284 

10.47371 

.93024 

13.33559 

0 

1 

.89576 

8.59332 

.91313 

10.51199 

.93053 

13.39547 

1 

2 

.89605 

8.62002 

.91342 

10.55052 

.93082 

13.45586 

2 

3 

.89634 

8.64687 

.91371 

10.58932 

.93111 

13.51676 

3 

4 

.89663 

8.67387 

.91400 

10.62837 

.93140 

13.57817 

4 

5 

.89692 

8.70103 

.91429 

10.66769 

.93169 

13.64011 

5 

6 

.89721 

8.72833 

.91458 

10.70728 

.93198 

13.70258 

6 

7 

.89750 

8.75579 

.91487 

10.74714 

.93227 

13.76558 

7 

8 

.89779 

8.7&341 

.91516 

10.78727 

.93257 

13.82913 

8 

9 

.89808 

8.81119 

.91545 

10.82768 

.93286 

13.89323 

9 

10 

.89836 

8.83912 

.91574 

10.86837 

.93315 

13.95788 

10 

11 

.89865 

8.86722 

.91603 

10.90934 

.93344 

14.02310 

11 

12 

.89894 

8.89547 

.91632 

10.95060 

.93373 

14.08890 

12 

13 

.89923 

8.92389 

.91661 

10.99214 

.93402 

14.15527 

13 

14 

.89952 

8.95248 

.91690 

11.03397 

.93431 

14.22223 

14 

15 

.89981 

8.98123 

.91719 

11.07610 

.93460 

14.28979 

15 

16 

.90010 

9.01015 

.91748 

11.11852 

.93489 

14.35795 

16 

17 

.90039 

9.03923 

.91777 

11.16125 

.93518 

14.42672 

17 

18 

.90068 

9.06849 

.91806 

11.20427 

.93547 

14.49611 

18 

19 

.90097 

9.09792 

.91835 

11.24761 

,33576 

14.56614 

19 

20 

.90126 

9.12752 

.91864 

11.29125 

.93605 

14.63679 

20 

21' 

.90155 

9.15730 

.91893 

11.33521 

.93634 

14.70810 

21 

22 

.90184 

9.18725 

.91922 

11.37948 

.93663 

14.78005 

22 

23 

.90213 

9.21739 

.91951 

11.42408 

.93692 

14.85268 

23 

24 

.90242 

9.24770 

.91980 

11.46900 

.93721 

14.92597 

24 

25 

.90271 

9.27819 

.92009 

11.51424 

.93750 

14.99995 

25 

26 

.90300 

9.30887 

.92038 

11.55982 

.93779 

15.07462 

26 

27 

.90329 

9.33973 

.92067 

11.60572 

.93808 

15.14999 

27 

28 

.90358 

9.37077 

.92096 

11.65197 

.93837 

15.22607 

28 

29 

.90386 

9.40201 

.92125 

11.69856 

.93866 

15.30287 

29 

30 

.90415 

9.43343 

.92154 

11.74550 

.93895 

15.38041 

30 

31 

.90444 

9.46505 

.92183 

11.79278 

.93924 

15.45869 

31 

32 

.90473 

9.49685 

.92212 

11.84042 

.93953 

15.53772 

32 

33 

.90502 

9.52886 

.92241 

11.88841 

.93982 

15.61751 

33 

34 

.90531 

9.56106 

.92270 

11.93677 

.94011 

15.69808 

34 

35 

.90560 

9.59346 

.92299 

11.98549 

.94040 

15.77944 

35 

36 

.90589 

9.62605 

.92328 

12.03458 

.94069 

15.86159 

36 

37 

.90618 

9.65885 

.92357 

12.08040 

.94098 

15.94456 

37 

38 

.90647 

9.69186 

.92386 

12.13388 

.94127 

16.02835 

38 

39 

.90676 

9.72507 

.92415 

12.18411 

.94156 

16.11297 

39 

40 

.90705 

9.75849 

.92444 

12.23472 

.94186 

16.19843 

40 

41 

.90734 

9.79212 

.92473 

12.28572 

.94215 

16.28476 

41 

42 

.90763 

9.82596 

.92502 

12.33712 

.94244 

16.37196 

42 

43 

.90792 

9.86001 

.92531 

12.38891 

.94273 

16.46005 

43 

44 

.90821 

9.89428 

.92560 

12.44112 

.94302 

16.54903 

44 

45 

.90850 

9.92877 

.92589 

12.49373 

.94331 

16.63893 

45 

46 

.90879 

9.96348 

.92618 

12.54676 

.94360 

16.72975 

46 

47 

.90908 

9.99841 

.92647 

12.60021 

.94389 

16.82152 

47 

48 

.90937 

10.03356 

.92676 

12.65408 

.94418 

16.91424 

48 

49 

.90966 

10.06894 

.92705 

12.70838 

.94447 

17.00794 

49 

50 

.90995 

10.10455 

.92734 

12.76312 

.94476 

17.10262 

50 

51 

.91024 

10.14039 

.92763 

12.81829 

.94505 

17.19830 

51 

52 

.91053 

10.17646 

.92792 

12.87391 

.94534 

17.29501 

52 

53 

.91082 

10.21277 

.92821 

12.92999 

.94563 

17.39274 

53 

54 

.91111 

10.24932 

.92850 

12.98651 

.94592 

17.49153 

54 

55 

.91140 

10.28610 

.92879 

13.04350 

.94621 

17.59139 

55 

56 

.91169 

10.32313 

.92908 

13.10096 

.94650 

17.69233 

56 

57 

.91197 

10.36040 

.92937 

13.15889 

.94679 

17.79438 

57 

58 

.91226 

10.39792 

.92966 

13.21730 

.94708 

17.89755 

58 

59 

.91255 

10.43569 

.92995 

13.27620 

.94737 

18.00185 

59 

60 

.91284 

10.47371 

.93024 

13.33559  1 

.94766 

18.10732 

60 

354 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


/ 

87° 

88° 

89° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.94766 

18.10732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440 

.98284 

57.26976 

1 

2 

.94825 

18.32182 

.96568 

28.13917 

.98313 

58.27431 

2 

3 

.94854 

18.43088 

.96597 

28.38812 

.98342 

59.31411 

3 

4 

.94883 

18.54119 

.96626 

28.64137 

.98371 

60.39105 

4 

5 

.94912 

18.65275 

.96655 

28.89903 

.98400 

61.50715 

5 

6 

.94941 

18.76560 

.96684. 

29.16120 

.98429 

62.66460 

6 

7 

.94970 

18.87976 

.96714 

29.42802 

.98458 

63.86572 

7 

8 

.94999 

18.99524 

.96743 

29.69960 

.98487 

65.11304 

8 

9 

.95028 

19.11208 

.96772 

29.97607 

.98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67.75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

69.16047 

11 

12 

.95115 

19.47093 

.96859 

30.83623 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.13366 

.98633 

72.14583 

13 

14 

.95173 

19.71737 

.96917 

31.43671 

.98662 

73.73586 

14 

15 

.95202 

19.84283 

.96946 

31.74554 

.98691 

75.39655 

15 

16 

.95231 

19.96982 

.96975 

32.06030 

.98720 

77.13274 

16 

17 

.95260 

20.09838 

.97004 

32.38118 

.98749 

78.94968 

17 

18 

.95289 

20.22852 

.97033 

32.70835 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

33.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

33.38232 

.98836 

84.94561 

20 

21 

.95377 

20.62876 

.97121 

33.72952 

.98866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

34.08380 

.98895 

89.46886 

22 

23 

.95435 

20.90409 

.97179 

34.44539 

.98924 

91.91387 

23 

24 

.95464 

21.04440 

.97208 

34.81452 

.98953 

94.49471 

24 

25 

.95493 

21.18653 

.97237 

35.19141 

.98982 

97.22303 

25 

26 

.95522 

21.33050 

.97266 

35.57633 

.99011 

100.1119 

26 

27 

.95551 

21.47635 

.97295 

35.96953 

.99040 

103.1757 

27 

28 

.95580 

21.62413 

.97324 

36.37127 

.99069 

106.4311 

28 

29 

.95609 

21.77386 

.97353 

36.78185 

.99098 

109.8966 

29 

30 

.95638 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

.95667 

22.07935 

.97411 

37.63068 

.99156 

117.5444 

31 

32 

.95696 

22.23520 

.97440 

38.06957 

.99186 

121.7780 

32 

33 

.95725 

22.39316 

.97470 

38.51855 

.99215 

126.3253 

33 

34 

.95754 

22.55329 

.97499 

38.97797 

.99244 

131.2223 

34 

35 

.95783 

22.71563 

.97528 

39.44820 

.99273 

136.5111 

35 

36 

.95812 

22.88022 

.97557 

39.92963 

.99302 

142.2406 

36 

37 

.95842 

23.04712 

.97586 

40.42266 

.99331 

148.4684 

37 

38 

.95871 

23.21637 

.97615 

40.92772 

.99360 

155.2623 

38 

39 

.95900 

23.38802 

.97644 

41.4452E 

.99389 

162.7083 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.99418 

170.8883 

40 

41 

.95958 

23.73873 

.97702 

42.51961 

.99447 

179.9350 

41 

42 

.95987 

23.91790 

.97731 

43.07746 

.99476 

189.9868 

42 

43 

.96016 

24.09969 

.97760 

43.64980 

.99505 

201.2212 

43 

44 

.96045 

24.28414 

.97789 

44.23720 

.99535 

213.8600 

44 

45 

.96074 

24.47134 

.97819 

44.84026 

.99564 

228.1839 

45 

46 

.96103 

24.66132 

.97848 

45.45963 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

47 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.43241 

.99680 

311.5230 

49 

50 

.96219 

25.45051 

.97964 

48.11406 

.99709 

342.7752 

50 

51 

.96248 

25.65546 

.97993 

48.82576 

.99738 

380.9723 

51 

52 

.96277 

25.86360 

.98022 

49.55840 

.99767 

428.7187 

52 

53 

.96307 

26.07503 

.98051 

50.31290 

.99796 

490.1070 

53 

54 

.96336 

26.28981 

.98080 

51.09027 

.99825 

571.9581 

54 

55 

.96365 

26.50804 

.98109 

51.89156 

.99855 

686.5496 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4369 

56 

57 

.96423 

26.95513 

.98168 

53.57046 

.99913 

1144.916 

57 

58 

.96452 

27.18417 

.98197 

54.45053 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98226 

55.35946 

.99971 

3436.747 

59 

60 

.96510 

27.65371 

.98255 

56.29869 

1.00000 

Infinite 

60 

XIV.—  TRANSITION-CURVE  COORDINATES. 


*. 

y  =  1C 

«=?(!  -J?) 

; 

y  —  ic 

*=m-j?) 

C  Dif. 

E  Dif. 

C   Dif. 

J5   Dif. 

0°10' 
20 
30 

0.00097  n" 
194  '' 
291  97 

o.ooooo  0 

10°  10" 
20 
80 

O.OoOOl  0~ 
5998  9' 
6091  5 

0.00314  n 
335  ]° 

40 

388  qi 

1   1 

40 

0190  Q 

346  j! 

50 

485  g 

o   * 
8   1 

50 

6286  ™ 

357  {} 

1 

0.00582  Q,v 

0.00003   * 

11 

0.06383  qr 

0.00368  t 

10 

679  Xi 

4   } 

10 

6479  2 

379  }| 

20 
30 

776  9' 
873  o,l 

7   * 

20 
30 

6575  ^ 
6671  \ 

391  jr 

402  }i 

40 

970  9< 

8  i 

40 

6767  J 

414  i5 

50 

1067  Ji 

10   | 

50 

6863  jj° 

4-26  J2 

2 

0.01164  OK 

0.00012   0 

12 

0.00959  q 

0.00438  , 

10 

1260  oS 

14   3 

10 

7055  JJ 

450  jo 

20 

1357  oX 

17  i 

20 

7151  \ 

462  Jo 

30 

1454  £ 

19  ; 

30 

7247  o« 

475  }o 

40 
50 

1551  g 
1648  g 

22   ;< 
24   3 

40 
50 

7439  9G. 

502  JJ 

3 

0.01745  Q~ 

0.00027 

13 

0.07535  q 

0.00514  .„ 

10 

18  12  Ji 

31   ! 

10 

7631  * 

5,»'~  J  *j 
2<   i.j 

20 

1939  Xi 

34   ! 

7727  ^ 

540  |! 

30 

2036  Xi 

37 

30 

7823  J 

554  Jo 

40 

2133  0- 

41 

40 

7919  ^ 

567  j^ 

50 

2229  g7 

45   J 

50 

8015  ™ 

4 

0.02326  ql 

0.00049   4 

14 

0.08110  Q 

0.00595  1f- 

}0 

2423  *X 

53 

10 

8206  96 

610  J^ 

20 

2520  Ji 

57    r 

20 

8302  9k 

624  J? 

30 

2617  £ 

62   2 

30 

8397  ^ 

639  J? 

40 

2714  9' 

66   * 

40 

8493  96 

653  i£ 

50 

2810  96 

71   g 

50 

8^88  ^ 

668  Jg 

5 

0.02907  ff. 

0.00076 

15 

0.08684  OA 

0.00683  ^ 

10 

3004  q,', 

81   J 

10 

8780  Q? 

6°8  jg 

20 

3101  Hi 

87   V 

20 

8875  ~ 

30 

3198  oA 

92   p 

30 

8970  *j 

729  ]5 

40 

3294  o?, 

98   K 

40 

9066  * 

745 

50 

3391  XX 
y  * 

104   § 

50 

9161     gg 

761  Jjj 

6 

0.03488  q. 

0.00110   R 

16 

0.09257  0 

0.00777  . 

10 
20 

3585  9' 
3681  X~ 

116 
122   S 

10 
20 

9352  ^ 
9447  9 

793 
810  1! 

30 

3778  X7 

129   ' 

30 

9543  •? 

826   ^ 

40 

3875  Xi 

135   |^ 

40 

9638  9^ 

843  Ji, 

50 

3971  $ 

142   J 

50 

9733  g 

860  g 

7 

0.04068  Q~ 

0.00149   „ 

17 

0.09828  Q. 

0.00877  <(, 

10 

4165  9' 

156   ' 

10 

9923  Q? 

894  JZ 

20 

4261  "X 

164   ° 

20 

10018  ^ 

911  }' 

30 

4358  97 

171   ' 

30 

10113  ^ 

929  } 

40 

4455  ,,;. 

179 

40 

10208  £? 

947  If 

50 

4551  jj° 

187   g 

50 

10303  g 

964  }£ 

8 
10 

0.04648  QR 
4744  X2 

0.00195   c 
203   i 

18 

10 

0.10398  „ 
10493  2'- 

0.00982  1Q 
1001 

20 

4841  Q' 

211 

20 

10588  ~ 

1019    {Q 

30 

4937  ™ 

2','0 

30 

10683  X? 

1038   ^ 

40 

5034  oA 

229   9 

40 

10778  95 

1056  '" 

50 

5130  o- 
y< 

237   | 

50 

10873  g 

1075  J9 

9 

0.05227  QR 

0.00246  10 

19 

0.10967  Qr 

0.01094  1Q 

10 

5323  oS 

256  ?2 

10 

11062  95 

11  13  4^ 

20 
30 

5420  5* 
5516  X 

265   " 
275  ! 

20 
30 

11157  *? 
11251  94 

1133  TQ 
1152  Art 

40 
50 

5612  S 
5709  X^ 

284   9 

2Q4   ' 
^*  in 

40 
50 

11346  95 
11440  ™ 

1172  *X 
1192  ~Q 

10 

5805  ,',,. 

304  j» 

20 

11535  jjg 

356        XIV.— TRANSITION-CURVE  COORDINATES. 


C  Dif. 


0.11723 
11912 
12101 


12477 

0.12665 
12853 
13040 
13228 
13415 
13602 

0.13789 
13975 
14162 
14348 
14534 
14720 

0.14905 
15091 
15276 
15461 
15645 
15830 

0.16014 
16198 
16382 
16565 
16749 
16932 


187 

186 
187 

186 


=  /(I  -  E) 


E  Dif. 


0.01252  4, 
1293  \\ 
1335  ;£ 


0.01464 
1509 
1554 
1599 

1646 
1693 

0.01740 
1789 
1838 
1887 
1937 
1988 

0.02040 
2092 
2144 
2198 
2252 
2307 

0.02362 
2418 
2474 
2531 
2589 
2648 
2707- 


« 


,Q 


39 

40 
41 

42 
43 
J44 
45 
4<i 
47 
48 
49 
50 


V-IC 


C  Dif. 


0.17388 
17661 
17934 
18206 
18478 

0.18749 
19019 
19288 
19557 
19826 
20094 


272 

271 


266 


20o2< 
20893 
21158  $ 
21423  2^ 
21686 


263 


22474 
22995 
23513 
24028 
24540 
0.25049 
25554 
26057 
26556 
27052 
27544 


c  =  1(1  -  E) 
Dif. 


E 


0.02797 
2888 
2981 
3075 
3170 

0.03267 
3365 
3464 
3565 
3667 

0.03876 
3983 
4090 
4199 
4310 

0.04535 


B001 


102 
104 
105 


111 

» 


246 


5739 
0.05995 
6256 
6523 
6794 
7070 
7352 


TABLE  XV.— DEFLECTION-ANGLES  FOR  TRANSITION-CURVES. 


Transit  at  P.T.C.,  n"  =  0. 

Tr.  at  quarter-point,  n"  =  \ 

/  £      0\                        1           A                        T> 

<°0    )    —     ~n~A0    -    -&0 
O 

(Si0)  =  jrfl  ~  B\ 

4>  for 

Bo  for  {  = 

£1  for  §  = 

n 

7, 

^0 

A 

^4i 

4                 ,•> 

n 

_L  —  1 

3     - 

40    Co      go    ]0o    joo    J4o    jgo 

4°  8°  12°  14°  16° 

.0 

.00 

.00 

.0625 

.0 

.05 

.0075 

.0025 

.0775 

.05 

.1 

.03 

.01 

.0975 

.1 

.15 

.0675 

.0225 

.1225 

.15 

.2 

.12 

.04 

.1525 

.2 

.25 

.1875 

.0625 

.1875 

.25 

.3 

.27 

.09 

.2275 

.3 

.35 

.3675 

.1225 

.2725 

.35 

.4 

.48 

.16 

.3225 

.4 

.45 

.6075 

.2025 

1    1 

.3775 

.45 

.5 

.75 

.25 

1    1    1 

.4375 

.5 

.55 

.9075 

.3025 

1123 

.5025 

.55 

.6 

1.08 

.36 

11234 

.5725 

\ 

.6 

.65 

1.2675 

4°25 

12357 

.6475 

1      1      2 

.65 

1.47 

.49 

1       1      3      5      7    11 

.7275 

1234 

.7 

.75 

1.6875 

.5625 

1       2      4      7    11     17 

.8125 

1346 

.75 

.8 

1.92 

.64 

1       3      6     11     17    25 

.9025 

1469 

.8 

.85 

2.1675 

.7225 

12      4      9    15    24    36 

.9975 

2      6    10    15 

.85 

.9 

2  43 

.81 

13      6     12    21     34    51 

1.0975 

3      9     15     22 

.9 

.95 

2.7075 

.9025 

1     4      9     17    30    47    71 

1.20*5 

1     4     14     22     33 

.95 

1. 

3. 

1. 

1     5    12    23    41     64    97 

1.3125 

1     6    80    31     47 

1. 

DEFLECTION- AKGLES   FOR  TRANSITION    CURVES.    357 
TABLE  XV.-DEFLECTION  ANGLES  FOR  TRANSITION  CURVES. 


Transit  at  mid-point,  n"  =  \. 

Tr.  at  three-quarter  point,  «"=£. 

<v>  =  7iV^ 

<V>=rV*i 

n 

</>  for 
j  j       .j 

Ai 

B.tar!f= 

A, 

*iHr  = 

n 

3   ~ 

5 

8°  12°  16° 

6°  10°  14°  16° 

.0 

.00 

.25 

1       1 

.5625 

1    4    11     17 

.0 

.05 

.0075 

.2775 

1 

.6025 

1     4     10     15 

.05 

.1 

.03 

.31 

1 

.6475 

1     3      8    12 

.1 

.15 

.0675 

.3475 

1 

.6975 

12      7    10 

.15 

.2 

.12 

.39 

.7525 

268 

.2 

25 

.1875 

.4375 

.8125 

1      4      6 

.25 

.3 

.27 

.49 

.8775 

"134 

.3 

.35 

.3675 

.5475 

.9475 

1      2      3 

.35 

.4 

.48 

.61 

0225 

1      2 

.4 

.45 

.6075 

.6775 

.1025 

1       1 

.45 

.5 

.75 

.75 

.1875 

1 

.5 

.55 

.9075 

.8275 

.2775 

.55 

.08 

.91 

.3725 

.6 

'65 

.  2675 

.9975 

.4725 

.65 

.7 

.47 

1.09 

.5775 

.7 

.75 

.6875 

1.1875 

1 

.6875 

.75 

.8 

.92 

.29 

1       1 

.8025 

.8 

.85 

2.1675 

.3975 

1      3 

.9225 

.85 

.9 

2.43 

.51 

1     2      5 

2.0475 

.9 

.95 

2.7075 

.6275 

1    3      8 

2.1775 

.95 

1. 

3. 

.75 

2    6    14 

2.3125 

1      1 

1. 

Transit  at  P.r.C.,,  or  P.C.,,  de- 

Transit at  P.O.,,  n"  =  1. 

flections  from    tangent  to  cir- 

cular curve. 

(8,°)  =  ^Cl,  -  B1 

70 

3 

(Sc°)  =  —Ac  -f  J?, 
a 

<f>  for 

B   for^  = 

B,  for  l£  = 

n 

f 
1 

AI 

1           3 

Ac 

n 

3 

4°  6°   8°  10°  12°  14°  16° 

4°  6°  8°  10°  12°  14°  16° 

.0 

.00 

1. 

1   5    12   23  41   64  97 

2. 

1    5    12   23   41    64    97 

.0 

.05 

.0075 

l'.0525 

1   4   11    20  36  58  86 

!9475 

1    4    11    20   36   58   86 

.05 

.1 

.03 

1.11 

1  4     9   18  32  51   76 

.89 

14      9   18   32   51    76 

.1 

.15 

.0675 

1.1725 

13     8   16  28  44  06 

.8275 

1    3      8    16   28   44   66 

.15 

.2 

.12 

1.24 

13     7   13  23  37  56 

.76 

13      7    13   23   37   56 

0 

.25 

.1875 

1.3125 

1   2     6   11   20  31   47 

.6875 

12      6    11    20   31    47 

25 

.3 

''7 

1.39 

12     5     9   16  26  39 

.61 

12      5     9   16   26   39 

3 

.35 

'.3675 

1.4725 

2     4     7   13  21   31 

.5275 

2      4     7   13   21    31 

!.35 

.4 

.48 

1.56 

1      3     6   10   16  24 

.44 

1      3     6   10    16   24 

.4 

.45 

.6075 

1.6525 

1248  12   18 

.3475 

1      2     4     8    12   18 

.45 

.5 

.75 

1.75 

1      2     3     6     9  14 

.25 

1      2     3     6     9   14 

.5 

.55 

.9075 

1.S525 

12469 

.1475 

12469 

.55 

.6 

.08 

1.96 

11346 

.04 

11346 

.6 

.65 

.2675 

2.0725 

1234 

.9:275 

1234 

.05 

.7 

.47 

2.19 

1     1     2 

.81 

1      1     2 

.7 

.75 

.6875 

2.3125 

1     1 

6875 

1      1 

.75 

.8 

.92 

2.44 

.56 

.8 

.85 

2.1675 

2.5725 

.4275 

.85 

.9 

2.43 

2.71 

.29 

.9 

.95 

2.7075 

2.8525 

.1475 

.95 

1. 

3. 

3. 

.00 

1. 

358         TABLE   XVI.— TRANSITION   CURVE   TABLE. 


CJrHJ>i-c     5O— OOiOrr     CO  CO  T  p  OO     Of  «O  —  t~  ••*«     O*  1-1  ~  <-*•?*     •«<  GO  CM  JO  4N     C:  O  -3-  :>?  CO 

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t-OOO5Oi-l     OJ^ttOCO' 

Ci(M(N(NCO     COCO'S'^iO     «5CO»t>l- 


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lOCOlOOS     COOO-^Ol-      lO-^-COCO 

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<i-Hi-H     WCJG<»OiCO     Tl<^tTj<lOO     t-t^OOO5O5     Oi-iC*CO-^f 


I1-*      1-H  1-1  1-1  T-C  TH      1-l(N(N(?iO} 


O  i-!  OJ  CO  •<*<     1OOOOO5< 


C5O5C5     O5O5O500OO 


38S3 


iioiOTfco  cocji—qp 

5C5O5C5O5  OO5C5OO5 

•  C^  i^  l^  C5  ^^  CO  O  i  -  O 

)  to  3S  n  to  «o  o  o  o  o 


W  Tf  «O  00  p     <JJ  "*  p  00  8     C*  TT  !O  CO  p 

ei  o»  c»  o< i  co    co  co  co'  co  i<    -<s<  rt<  •*  -^  o 


S3888 


rrtOOOO     T^  — 1<  O  OO  O     CfTOOOO     O^ftOOOO     C^'TCOOCO     O1^1 


TABLE  XVI.— TRANSITION   CURVE   TABLE.         359 


loooo  ooooo  ooooo  ooooo  oooo< 

-       -8*1  04-0000  g*3££o  g-ggcoG  c>5»oo< 


S8S8 

co  co  O  l- 


coco  e*oo 


i  T-I  T-I  I-H    T-I  o*  o»  oj  o<    cocoww    **  in  in  eo  co    CD  <>  t-  GO  GO 


Ico^o    8°-c 


>  OS  OS     OsOiGOQOGO     t~  t-  I-  CO  CO     -O  in  TT  CO  CO     OJ  — i  O  O  O5 


ISSSS 


fei  ~  '  '^«  T-; 


t>t~GOO5     OlOi-iCJOO     OOTfiCCOJ- 


•  1010     COCOt-t-GO 


~,-<T-iT-io*    cxMcocoeo 


OOi-QO  OS  ( 

T-J    woicoTr'io    t^odoii-^co    TTCOGOOI 


^888 


C<  OJ  C*  (?J  < 


ScScSc^cS    55 


•<*  o  in  o  in 

i^^    IcoIninS    SSccccS 


8S8S  8S8S8  S8S8S  8S8S8  S8S8S  8S8S8  S8°SS 


CO  CO  r»<  Tfi  irt 


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T-I,-,      ,_,,-,  T-,  o  CN}      OJO<COOO< 


§0000   o 
i-  oo  os  55    -H 


O5O5     O5O5O5O5O5     GOGOOOGOt--     t-COCOCDlO      inTfTfW?* 

-coos    §  JJ  J|  (j,  w    cvojc-lcis*    cococbcoco 


T-,    T-,  ^^ ,_,  c*  c\j    ojcocoTrTJ1    inococot-    t-cooioso    T-I  I-H  c*  so  ^< 


3  T-S  T^  o*    w  cv  N :  cxi  co    co  rf  m  in  co 

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10  CD' {.- oi  o'     T-I  TO  to  CO  00 


00  CO  0  l- 


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ir^i-i    oJoicJcJcN*    cococococo    5TT"o<-^T(<    loinininm  eoococo! 


oooo   ooooo   ooooo   ooooo   ooooo 
GOO* So    ^GOOJCSO    ^co5jcoO    ^-cooioo    ^ooolSo 


360         TABLE  XVI.— TRANSITION  CURVE  TABLE. 


11 


>ooo   ooooo 
sioioco    cb  o  «o  ts  «.- 


CO  C5  <?J  CO  O    TF 

ci  ct  co  cc i  •*'   -iji 


5O5O50O  OO  i^  l~  I-  CO  1O  •>*•••*  CO  SJ  i—  O  O5  GO  CO  1O  T)«  CO  i-  OS 


Ilgl§  islil 


JOCGp  OC  1^ 


lO<OCOi>-00     OOOOi-i5>»     COTPlOi~GO 


SNOIW 

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>  TI  •*?  £-  m 

T-ri-(T-tO* 


mcoi>ooc5   Oi-i( 


T*Ct     C*CO1OCOOO     OS  1-1 


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lOiOlOCOOO     OSOO^-i-i 


T-I  »-i  1-1  IN  (N  COCO- 


•  CO  GO  ^ 'I"  GC  CO  C 

•  CO  OS  Tt-  o  i-  CO  < 

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83S8 


i-ii-icveo    coTfrj-ioco    «ot-t-ooo5    0300  —  0*    o»roco^»o    iococot-oo 


TABLE   XVI.— TRANSITION   CURVE   TABLE.         361 


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1  T  CO  GO  O  <7J  Tp  CO  X)  O  O>  "^  CO  CO  O   C>  -^  CO  C/v  O   CJ  Tf  CO  CX  O   C"?  T  CO  OO  O  C?  — "  C*;  GC  O 


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>OOO?      i— (1OCOOCO     lOi— *^t-t*     C?  i— i  1O  •**  1>-     T*  CD  CO  *P  i— t     T— (  CD  IP  OO  lO 

•ooeo    oococsow    ost-»ocow    cjciwco-^    ooor-.Tfioo    (NO^OOJ 
<  1-1  o<   e*  03  TO  T)«  10   10  o  t-  oo  os   o'  r-'  oj  co'  TP    10  o'  oc"  os  d    ci  co  10'  d  oo 


ot-ocoo 


asosooc-o 


ioco»-iaoio 


OOco^rC^O    ODOTfOJO    OCOTt*^O    OOO^C^O    OO§^^O     OOCD^rC^O 

pi-^f    -^i»n'oi>oo    ooosdr-J'TJ    w'coTt<»od    di^'ooosd    o'i-«ci' 

1— I  TH   T— I        1— I   ,— I  fH  l-«  TH        1-^T-H-I— «^N(7?        T.'7?C1 


362         TABLE   XVI.— TRANSITION   CURVE   TABLE. 


ill  §1111  Hill 


c»  t-  oo  cs  o»    o^ 


OS  OS  OS     OO  OO  t-  I-  «O 

OO      OOO5O3O5      OJC5O5O5O5 
TfAO     «Oc-l-OOOi     O  —  IWCOTJ< 


OSt-lOCOi-l     OSCO 


•  %&%% 


TTT-.QO     lOi-it-CCOS 


i-i-TT>CO 

weoTrio 


oeo       co 


ooocoo   ^oco^o 
T-J  <M'  oi  oj    o«  so'  TO  cc  TJ< 


TfTflrtCOCO     £>t^OOO5Oi 


TI  ci  oo  •<*  «d  oo  o  ffj  »n  oo  i-<  T»<  06  o>  co  "  in  o  in  T-<  o  o>  06  10  »H  od  m'  oi  oi  i-  in  co  ^-< 

»-.i-i     T-i^Wff^iN     COCOrtTriO     lO«DOl-t-    OOCSOSO-.     T-.0>COTfiO 


>S     TJSO-^IIO    COJ>COOST-I    (Nco-^ioto    i-  oo  o» 


o  o?    co  *^"  iO  co  r-    oo os o  —  co    Tf*o?Of^co 

OJW     CiWCJWW     OiWCOJOSO     SOCOCOCOW 


o^!o*co    oSosr-i-*    t-o-*co^    i-T^5o_e^i~    coosio'wos    S ^ T-I §5 S    SOTJ-CO' 
T— ITH    T-iojojoico'    CO^TJ<AOIO    cooi'-oooo'    os'o'T-«-f-IcV   CO'TP»OCO'I 


(NOQOOCO 


OOOO    oooiosoJ    oioioJoioJ    os'oJoiOioi    oioooooo 

WOOTTJO     «Ol-i-OOOJ     O--i?<JOTl<     lf5«5l-OOO5     O  T-I  ?>  CC 


WCOl-CO     1OCOCO5OT-H 
r-NTl-     OTTOOCOOS 


53?S8S§  55feS!gS  J?g^38  K; 


T-    l-H        T-     T-    T-     Ot  Cl        Cl  W  O      «  CO 

^,  ^  o  ^ 


IO  eO  CO  t-  eO     -^i-i 


8888  88888  88888  88888  88888  88888  88888 

^O     COL-QOOSO     -NCOO  » 


O     CJ^CDOOO     C^*^OGCO     C^"rf^ODO     T^T'^DCXDO     C>TCOGCO     O^tOV'O 


TABLE   XVI.— TRANSITION    CURVE   TABLE.          363 


8  §ffs§  !l|||  Hill  §1111  islll  lllll 


r-n-n-(  C  IN  CO  CO  T)«  Tf  in  O  ?O  1-  GO  00  OS  O  T-I  <W  CO  TT  1C  tO  l>  GO  OS  O  r-I 


COOJOQCO 


:s3&S 


i-  f-  <M  oc  'CO    ^  i    co  i   0     oa  o  c  o    iO    so  oo  o      c 


O^ODCOQO 

i  i-i  i-<  T-C  <j«    oiwojeood    T^  Tt  -<t  m"  »o 


wca. 


OOCiOi     C5O5C5OiCi     QO 

^.»i-«  -««« 


§§8S  SSSSS  SS88S? 

AO  o   i>  oj  o  i-<  co    TJ<  ic  o  06  oJ 


SSSSS  SSSS  SSSSS 


JCO      TP  TP  T  •*  T 


5  lO  I-H  T-I  rf     OOCOOO     CO  O5  Oi  <JJ  t^-    I-  C5  lO  Tf  10     OSl-CD'-t-     COOOCOOO 
5  GO  1-1  ^  I-     i-llCOS'S-Ol     TfOSlOCJGO     lOWOCOCO     TfCCWO»T-i     i-iT-(?^COrr 

'THTH'TH  oi  oi  OJ  CO  CO  •*' •*  10' <O  O  t-' 00  OS  OS  O  T-I*icOTr'm'  «d  t-  CO  OS  O 


T)«COi-"OQO 


00  -^  T-I    IXN 


osco^f^^co    j^-iooc^o    ^i 

^«»««  ^irJio-coV  w^Vwog  £'S-S'25J  «^^'^o  coVVd 


O<NTJ»?OOO   owmcoi-i 

,_;,_;,_;,_;,_;  oicjjNoico 


t-     00000=0- 


i-l     QOJOi-HCOO     -*J>05< 
OJC^OJC^C^     COCOCOC 


coqoOGO   Tj«o50>eo! 

lOAOlOlOiO     1CCDOSC* 


-TOCC'O     C-^TfOGC'O      C^^TO^jO 


>cO;O         > 


364         TABLE   XVI.— TRANSITION   CURVE   TABLE. 


>  rr^  -*i*  z3 
I  CJ  C<  Oi 


i  5J  ?J  ?5  M 


IS*     CC  CO  CO< 


§§l^^^ 


_"*t-    00  Win  OS  CO    t--r-«D*« 
r-I  i—     i-t  CJ  <N  d  CO     CO  rji  tli  10 


OO  OS 

OOOSOT-KN    SO-<*OI>-QO 


>OTH 
i  W  O* 


CiQO     t>iCO*C»iQ     OCO^OOQO     t-  ^  O 


CO  i-t  t*  i-(  Ol     *-»  00  I-H  C*  O 


OJO5GOQO    t- 


|T-<i>.(jjt~    (Ncopeoo 


^5?jcoeoco    e§°^^^ 


CO?>OOp(N     TttOQOi-cr}<    t-pT)<QO(N     OTHIOI-HI.-- 
THTH     THTHrHC*S\f     (NeOCOCO''*     «4*  to  id'  CO  CD 


TH  ci  co  ^   m  to  oo  T-I  co 


CDt-COOOO5     O  TH  SO  ^> 


TH  OOCOO 


gfcSSS     SS?^' 


.T-co^eo    060  ~ni 

iiCAO4O*O     iOCOOO1; 


OOWOO     T»tQO(N«00     TJ«CO(N«00     ^OOCJCOO     TfOOINOO     •'S' 


oaoo    OT^CQC 


-^'tciaoo    ^*t<coGCO    c^^ccooo    OT^SOOOO 


TABLE    XVI.— TRANSITION    CURVE    TABLE.          365 


'™     OT?  00  coop    5£~ 

>'  o   »-«'  i-i  TH  cv  oi   co  •* 


8SSSS  SS8S£8  53S8S3  Sii^gS 
tJcdos-o-'  s«s»V  SSS?i?5  8SSS5 


01    ojcocot-o 


o«o«co    eo  t-  1-1  »o  oo 


t-oo-o       - 


SSSS15 


•»—  d     CDOiO^?O     i—  t- 
O  T-i^OJ5O     Tf 

I-H     T-(  i—  1-1  C«  7»     COOOCOrfO 
i—  <     (MCO^TCOQO     OCOOO»O 

o  T-i  f-i  »i  a*  w    coTfi-<*inco    t>t^oooi-H 


OOl-t-ODO 


OSOOl-     JO  91  Op  OI  CO     OOQOt-rt<O5     WWOOCS     OGOOlT^'N     OOOGOCOO     W5O»! 


0000  OOOOO  OOOOO  00000  00000  OOOOO  OOOOO 


i— i  O     TfSOl^lOGO     COQClO^OC*      C>CD~*CO'7>      f— 1  ^  O  O  TO      OOlOlOS^ 
r-tO      1-,1-SOOl--       JOSO(Nl-.rH       ^-lr-.(NSOJO       t      *  W  55  O3      8?  25  Q  3>  g) 

T-<i?»ci   so  so  rf  jo  10   cot- GO  oio   T-jwsoTfio   ot-oso'i-i    co'-teoi-cd 


i  CO  »-i  OS     r-  T}<  TH  CO  •<«•     O»Oi-i»OO 

' 


r-    w<»« 


)  so  o  10    co  ^  t^  T— '  t«- 

i-iT-iiT*     CO  ^|  iO  £»  GO     i-HSOlOOpT-i     TTGOWOi-i     OG*GO-^'i-i  OOeDTj«SOC* 

T-H i-i i-J I-H oi    cNJcisoeo-*   rfjo'ibcoi-  j>Goosdi-i 

W     WSOSOTtiiO     lOCOt-GOOS     O»-<5*SO»O  COt-OSi-KM 

r-^THi-li-(i-(  i-i  T-H  i-^  O?  »>' 


-»•'  o'  «o    CJOJOCO'T-^    O5j-«dirJin    -^io'ic»ca3 

SO  TP  TJ<     lOOCOl-QO     QOOSOi-M?>     CO-riOCDl- 


:  r-i  55  5} 

•  ?)  CJ  ?> 


OS  OS  GO     OSOO5OO     TTOt-CO"*     Orj<l-t-io     i-nCOTfO     Tj<T}<C?l>OS     OOrfCDiC- 


-<rOGOO     OfTfOCOO 


f 


366          TABLE    XVI.— TRANSITION    (TL'VE   TABLE. 


OO 


3 53 SS  85  SSSSS  JSS8S3  8S3&SS 


C5  T-I  i-l  i-l  O  Ci  CO  CO  OS 


co  co  10  rrcow^j^ 
<-  oo  os  o  i-i  o»  co  I* 


TJ<  O-  0  W  CO 
CV  <TJ  O*     CN>COCOCOCO 


OS  CO  GO  rf  T-I 


>COJ>OJt~     00  OS  «0-<*  CO 
y-^~l      .-.CJWCOSO      T*  Tj«°  O  CO"  t>     OOOsdi-^O*      COJOCOGOO 

'incot-oo   osowcoo    «oooo(?*Tt<   oooo(NTt< 


'  i— '     OSt'-COCOCO     COt'-OOOii— *     COirjon^-i-H*     f~  i— J  »n  'yi  ^> 
.CO     COOOOOsS     ^^«^«     ^2S?»^     B'^if^S 


<N  00  C»  10  to     Tt<  O  • 


-* oo 01  coo   TJ-C 


•oowcoo    ^oorjco< 


!-!»-<(?« woo    eo'^irjsot^    i- coos'-- isi    eoTj.'»n'i-od 


OS     OSGOt^-COiO     COi^OSCOCO     OCO3^t~TH 
>OO5     OSOSO5CJCS     CSOJOpOOGO     GOl-t-COCD 


SS^KS  SScSSS 


»  O5  CO  t—  i— '     CO  GX  CO  lO  W 

i  i-i  cv  c*  co   co'  rii  Tf  «n  co' 


>j>c-i>.   oppi?»»op 

•'cdos'o    i-so^t»o'j> 


•OiOOS^i     TH 


C7-S-COOOO     0>T)«COCOO     OjTtCOOOO     ClrfCOCOO     WTfcOOOO     U?Tt<COOOO 


)CC^     CO  1O  t- 


TABLE   XVI.— TRANSITION   CURVE   TABLE.          367 


?t  5f  f<  ?»  5  T-<  ?>  rj  ?>  7*  co  co  £r  so  eo  co  J<: 


Z%  lo-SSSS 


TH    T-I  T-I  r-i  T-I  o»    !W7*sococo    -<i<  T  »n  in  o    o  t~  t-  x  GO    os  os  o  ^  T-I  < 


p  OO  00  l>  t-  p  p  >n  Tf  CO  <N_  TH  CS  00  p  •«#  <??  O  OO  CO  CO  T-I  X  O  T-  X>  ^TI  O 


/-50  223SSSS  SS2SSS  S5®  =  5r  ° 

'THT-ir-ioi    CNJCNteoso'-^    10 »o cot- 06    xosdi-lci    eo"-* 


^-i-HTHT-iN    oJcioicoeocoTj! 


01  oo-*  10  co 


— i  CM    ojcomcooo    os  T-I  co  10  x> 


OS  OS  QOOO  t-lO  •* 


CO  l>  00  OS  CO 


i-  p  s*  so  w  »-  p 

C-5  CO  CO  CO  CO  CO  CO 


n- oo  woo  rr-( 


»O  £•-  CD  OS  r-i  W 


c  o>  o  o  i-  « 


13535    osaocct-t^   co«o»o-yeo    WT^C 


TT  0*  OS  l>  Tf  1-1  CO 


sss 


!  W  CO  CO  -^     -^  O  CO  5O  t-     00  Os"  OS  O  rH      CJ  CO 


gTHC*        01  < 


T-I        TlT-T-lT-lT-.        <NCJCN((NOJeOCO 

^HT-I    O}o»o?(7»o>    eococOTi'-s'    lomcocot-t-t- 


T-I-I-I     W  SO  •»*  O  I-     OS  O  W  -t  O     OS  »-i 


•^H     •^-UCCO-'OS     t^  I-  O  O  lO     OCC 

x    oo  cs  <?«  o  T-I    oo  o  p  p  oo    TH  m 

sU&gusi  ^sfess  gj: 


5     CSOOl^plO     00  T-I  p  p  CO     plOOlOOS     COOOOOTH     -riOOS) 

§^    SSScccs    S2?jco5§ 


o  10  p  eo  «o  t-  oo 


t-'  os  d  I-H  co    -i« »!?  o"  x  os'   d  o?  co  -r « 


1 10    oo T—  -^ t- o    cocoo5?>inxi-i 


>o    oooooo 

:  O     '— '  ^'  CO  T  1-7  CC 


368         TABLE   XVI.— TRANSITION    CURVE   TABLE. 


8SS83  2 


'COCO     COt-OlOTJ     O  O  O  O*  CO  ~ 

;I-HI-    oo  os  coo*  os    <5  «5  5  £9  at 

m  co  co  t- 1-   06  oo  os  o  o    «'  o»  co  co  •<*  in 


O5O5O5QO     GOl-CDin^     COWOOSt-     iOWOJ>-iO     1-1  00  r»<  1-1 1>     OJ  90  CO  OD  OJ  CO 


lOOiO     OTCS'-fOS     —F  OS  •**  OS  Tt 
.-(C*C<     COCOCOrf^'     10  10  COCO  I- 


~  10  O?  -f  O  < 


co  co  T*  o  o    co  i>  ao  os  o    t-crjco^co 


inpt-osp    <NTf«oacp    co  co  os  o»  10    oscoi>i-(co~ 
i-H    ,-i  i-c  i-i  r^i  ei    CN|  w  o»  co'  co    c<: 


OSCOCO     t-^prTTr 
<  7J     CO  10'  CO  OO  O 


)  os    op  t~  in  co  p    t>  co  00  < 

>O5     OSOSO5OSOS     OOOOt^-i 
)^i     OCOJ>Q005     0-H7J< 


p  ?i  co  co  3»  p  i-  oo  t~  p 


'  1-1  GO  -*  OS  O*  OO  <N  O  t- 

joco 

'  CO  TO 


I  •— i  O5  CO  "^  »—  < 
1  lO  lO  CO  t-.  QO 
!7>7J?JOJ  Oti 


!0     COCOrfWO     OOCOTSJO 


OOpTj<C?p     GOCOTTWO     QOp- 

«--  os  i-I  co  ITS    CD'  oc  o  ^  •*    in'«-'c 
coco^Tr1^    •^•^lOino    lOoi 


oooos    ct  t-  rf  TJ<  «o    oscoTfmoo    cooosoco    aoin-tmt-    ojoocomco    ooi'?i~-^">>-> 
rl  £i  et    4f  10 1-  os  1-1    co  co  os  7*  10    os  co  co  I-H  10    os  -*  os  TT  p    in  p  p  7}  oo    ^  ^  t-  ^_  ^  dp 

'  r^   ,_;  ^J  rt  cj  <jj    oj  co  co  Tji  -^    Tr  in  o'  co  co   i^  cc  oo'  os'  os'   c>  i-<  i-i  c*  co'  '.o' 


O5OSO5OS     QOOOt^piO     TjiCOCJ-iOT     Oppr»<O?p     r-iOG>*p»O     COOO^pini-i 

'35    rr  os  Tf  os  •*    os  rt<  os  MH  oo'    co  06  co  oo'  co'    t-  cj  t-  i-c  co'    ~  in  o  o  os'  •*' 
*    inincocoi-    t>-ocoDO5os    o  o  1-1  -<  w    !?j7Oco-^<Tr    mincococot- 


*-it>-OC     TfTfOSOOC^     O  -i*  ^— •  CO  O     •— •  Is-  CD  i— *  O     C7OC>>O5OS  C^O^T-HO  CDlP{-—^rTfOD 

wcoio   »^5So5«   opcocisio?*   ospT»|coo*    ~,;=;.-,rto*  T»-CDOO^ITK  i-^lnp>np 

i-Hf-ti-HCV     GMCOCO^*lO     lO  CO  t—  00  OS     O  T— '  O^  CO  Tf  1OCO£'-OSO  i— '  CO  ^  CO  l^-  O5 

^H ri  1-1  ^i I-H  i-i;=;;_rtcw  c.-  -.>  ~>  ~>  ~>  -.< 


0>i-H!NTjHp      00  0  CO  000      r-l  T)<  t- TH  10  OS 

Otcocoeo-^    T^^OOCO    coi-ooaooso 


)  Tf  10  t-  05      —  ( 


I  !-(        O    »-H  OO  «O  p 

33^SS  SSSSS  S8S88S 


S  OS     Op  t^-  CO  •*  Q* 

O  OS  OS     O5  OS  OS  OS  OS 
C070-*     OCOl-SOOS 


'i-i    ppcoop    iOTj<i-ippco 


cocjaoTfo    COC*OOT)«O    -^wao' 


co  c*  oo  -rr  o  co 


S  2 


TABLE   XVI.— TRANSITION    CURVE   TABLE.          369 


Si~  co  t~  t-i  GO    t-  oo  -r-1  co  co    o*  <?*  jo  oo  co    ost-jo- 
C5  T  O5  lO  O     CO  ??  Oi  1C  W     C5COCOOGC     JOCOr^l 


i-HT-ii-i    T-HOJWCOCO    CO-O«TJ<JOCO    CO  ll- 


s  o  i-  o  a*    co  "•»  »o  10  co 


CS  050  GOt-     !>  JO  JO  COW     OGOCOCOO     t- "*  O  CO  O*     GO  CO  GO  M  CO     Cl  W  JO  3D  O  1-1 


,-  ^  5,1  Oj  CO 


ssssss  gs 


p   QO^IT)<COT-I    iOOieoooco    os  o  TI oo «n oo 

rt     T-I  OJ  ffi  CTi  CO     CO  CO  •<*  TJ<  10     ITS  »  1--  I-  00  OS 


5OGOO  O?  Tf" 


feSSSSB 


O5O5GO     I-  1O  Si  OS  JO     pJOGOOi-l     -^Oapr-iJO 


CO  05  CO  TT  COO 


•  OO  O  0*  •*  p  GO  p  O*  rf  p  GO  O  OJ  •*  p  GO  O  OJ  •*  p  GO  O  CJ  Tf  p  OO  O  <N  ••*  p  QO  O  < 

~-*>    cdaoo-(Mjd    t^os-i—coco    ODO'C^TPI-'I 
*  ^    TT  TP  jo  jo  jo    jo  10  co  co  co    co  t~*  t^- 1^- 1~  i 


'r-li-i     r-i  W  0>  G-i  SO      CO^^'oo     Co'dl-^t-GO      OSOSO'-!^'     (N  CO -f' TT  JO  CO 


os  os  os  co  co    t-t-jo^t-co    <wpcopTj<    ^HCOJOOJOS    lOr-cpwt-    ^-ppcopos 

ife^1-^  8«3SS5  SSSSS'g 


'  i-  o  rr  oo  co  ao  JO  I-H  oo  p  TT  ec  co  w  cc  •v  jo  i 

^,-i^OJi;^     CO-iJ'-'S'idcO     l>a6oiO'rt     o»< 


S    Wini'qS    SSJcB^S    cocoo-eo 
ST-WCO 


co-Npp-*    oscooo^pp 
oicdco'^-^'    TTJOJOcdt^i> 


!co^    SSScocj    o^VTcc 


poo   pooop   opppp   opppp   opppp   opppp   pppppp 
^   Is&aoo    d^cocoo    w^cpaoo    g^^^aoo    ^^»^o 


CO  CO  CO  CO  ( 


370         TABLE   XVI.— TRANSITION    CURVE  TABLE. 


J>5O   10  •<&  yt  r+  QO   «5«o 

S8SS5  S 


ass 


Jt--    iowoioos    coo»nTt<cj    QOi-ieo 
osoJoi    oJoiooooi^    t-^o'o'^'eo    —'006 

77IOTT     U7«5l-Q005     O^T.»CO"*     100«0 


OSCRCOOOt--     O»OCCCMO     CDOCO 


Ttf^Ol—     O  O  O?  1-1 1~     C5  QO  CO  iO -f     CiOt- 

»«£co   SRtvwoo^   i-.oi:e<-t-   i-oso 

,-!  i-i  C*  CS>  CO     Tl<  T»<  O  so'  i-     00  O5  i— ' 


eicm 


rfiQOd«DO     ^  CO  W 


SSS88  S 


1-1   i-  <N  o*  eo 


»-iTHCNC*e»   cc 


OJQOl-     Tf^i 

siosoJ    o^osoOGOt-    ",c  o  -i'  o  i- 

- 


-    ",        -'      i-          ~  -or 

1-  OO  Ol     O  1-1  C>  W  -^f     T  »O  ffl 


ooo   ooooo   opqpo   pop 

JIO     GO  rH  "^  I—  O     COOCSC^IO     CO  i— '  ^ 
,,_     T-,c>C\»OiCO     COCOCOTTTfi      TTilJlO 


i-i  <w  o  i-  os  <N  4>  OJ  TP  oo  co  QO  co 
'  i-i  i-I  i-i  oi  o»  05  eo  ^< 


OJOSCOt-O     OCOi-(O5L-    Tji^ 


OJOOt-     K5S»OOCO«O     O5OOS5OC*     W5  «O  «O 


OO«Orf(NO    OOOrl" 


SRS 


TABLE  XVII.— AREAS  OF  LEVEL  SECTIONS.       371 

Base,  2b  =  14  feet.     Side  slopes  1^  to  1. 


C.  H. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

1.4 

2.9 

4.3 

5.8 

7.4 

8.9 

10.5 

12.2 

13.8 

1 

15.5 

17.2 

19.0 

20.  r 

22.5 

24.4 

26.2 

28.1 

30.1 

32  0 

2 

34.0 

30.0 

38.1 

40.1 

42.2 

44.4 

46.5 

48.7 

51.0 

53.2 

3 

55.5 

57.8 

60.2 

62  5 

64.9 

67.4 

69.8 

72.3 

74.9 

77.4 

4 

80.0 

82.6 

85.3 

87.9 

90.6 

93.4 

96.1 

98.9 

101.8 

104.6 

5 

107.5 

110.4 

113.4 

116.3 

119.3 

122  4 

125.4 

128.5 

131.7 

134.8 

6 

138.0 

141.2 

144.5 

147.7 

151.0 

154.4 

157.7 

161.1 

164.6 

168.0 

7 

171.5 

175.0 

178.6 

182.1 

185.7 

189.4 

193.0 

196.7 

200.5 

204.2 

8 

208.0 

211.8 

215.7 

219.5 

223.4 

227.4 

231.3 

235.3 

239.4 

243.4 

9 

247.5 

251.6 

255.8 

259.9 

264.1 

268.4 

272.6 

276.9 

281.3 

285.6 

10 

290.0 

294.4 

298.9 

303.3 

307.8 

312.4 

316.9 

321  7 

326.2 

330.8 

11 

335.5 

340.2 

345.0 

319.7 

354.5 

8:>9.4 

364.2 

369.1 

374.1 

379.0 

12 

384.0 

389.0 

394.1 

399.1 

404.2 

409.4 

414.5 

419.7 

425.0 

430.2 

13 

435.5 

440.8 

440.2 

451.5 

456.9 

462.4 

407.8 

473.3 

478.9 

484.4 

14 

490.0 

495.6 

501.3 

506.9 

512.6 

518.4 

524.1 

529.9 

535.8 

541.6 

15 

547.!: 

553.4 

559.4 

565.3 

571.3 

577.4 

583.4 

589.5 

595.7 

601.8 

16 

608.0 

814.2 

620.5 

626.7 

633.0 

639.4 

645.7 

652.1 

658.6 

605.0 

17 

671.5 

678.0 

684.6 

691.1 

697.7 

704.4 

711.0 

717.7 

724.5 

731.2 

18 

738.0 

744.8 

751.7 

758.5 

765.4 

772.4 

779.3 

786.3 

793.4 

800.4 

19 

807.5 

814.6 

821.8 

828.9 

836.1 

843.4 

850.6 

857.9 

865.3 

872.6 

•20 

880.0 

887.4 

894.9 

902.3 

909.8 

917.4 

924.9 

932.5 

940.2 

947.8 

21 

955.5 

903.2 

971.0 

978  7 

986.5 

994.4 

1002.2 

1010.1 

1018.1 

1026.0 

22 

1034.0 

1042.0 

1050.1 

1058.1 

1006.2 

1074.4 

1082.5 

1090.7 

1099  0 

1107.2 

23 

1115.5 

1123.8 

1132.2 

1140.5 

1148.9 

1157.4 

1105.8 

1174.3 

1182.9 

1191.4 

24 

1200.0 

1208  6 

1217.3 

1225.9 

1234.6 

1243.4 

1252.1 

1260.9 

1269.8 

1278.6 

25 

1287.5 

1296.4 

1305.4 

1314.3 

1323.3 

1332.4 

1341.4 

1350.5 

1359.7 

1368.8 

26 

1378.0 

1387.2 

1396.5 

1405.7 

1415.0 

1424.4 

'1433.7 

1443.1 

1452.6 

1462.0 

27 

1471.5 

1481.0 

1490  6 

1500.1 

1509.7 

1519.4 

1529.0 

1538.7 

1548.5 

1558.2 

28 

1568*0 

1577.8 

1587.7 

1597.5 

1607.4 

1617  4 

1627.3 

1637.3 

1647.4 

1657.4 

29 

16G7.5 

1677.6 

1687.8 

1697.9 

1708.1 

1718.4 

1'728.6 

1738.9 

1749.3 

1759.6 

Base,  26  =  15  feet.    Side  slopes  1^  to  1. 


C.  H. 

.0 

1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

o 

0.0 

1.5 

3.1 

4.6 

6.2 

7.9 

9.5 

11.2 

13.0 

14.7 

1 

16  5 

18.3 

20.2 

22.0 

23.9 

25.9 

27.8 

29.8 

31.9 

33.9 

2 

36.0 

38.1 

40.3 

42.4 

44.6 

46.9 

49.1 

51.4 

53.8 

56.1 

3 

58.5 

60.9 

63.4 

65.8 

68.3 

70  9 

73.4 

70.0 

78.7 

81.3 

4 

84.0 

86.7 

89.5 

92.2 

95.0 

97.9 

100.7 

103.6 

106.6 

109.5 

5 

112.5 

115.5 

118.6 

121.6 

124.7 

127.9 

131.0 

134.2 

137.5 

140.7 

6 

144.0 

147.3 

150.7 

154.0 

157.4 

100.9 

164.3 

167.8 

171.4 

174.9 

7 

178  5 

182.1 

185.8 

189.4 

193.1 

196.9 

200.6 

204.4 

20S.3 

212.1 

8 

216.0 

219.9 

223.9 

227.8 

231.8 

235.9 

239.9 

244.0 

248.2 

252.3 

9 

256.5 

260.7 

265.0 

269.2 

278.5 

277.9 

282.2 

286.6 

291.1 

295.5 

10 

300.0 

304.5 

309.1 

313.6 

318.2 

322.9 

327.5 

332.2 

337.0 

341.7 

11 

346.5 

351.3 

356.2 

361.0 

365.9 

370  9 

375.8 

380.8 

385.9 

390.9 

12 

396.0 

401.1 

406.3 

411.4 

416.6 

421.9 

427.1 

432.4 

437.8 

443.1 

13 

448.5 

453.9 

459.4 

464.8 

470.3 

475.9 

481.4 

487.0 

492.7 

498.3 

14 

504.0 

509.7 

515.5 

521.2 

527.0 

532.9 

538.7 

544.6 

550.6 

556.5 

15 

562.5 

568.5 

574.6 

580.6 

586.7 

592.9 

599.0 

605.2 

611.5 

617  7 

16 

624.0 

630.3 

636.7 

643.0 

649.4 

655.9 

662.3 

668.8 

675.4 

681.9 

17 

688  5 

695.1 

701.8 

708.4 

715.1 

721.9 

728.6 

735.4 

742.3 

749.1 

18 

756.0 

762.9 

769.9 

776  8 

783.8 

790  9 

797.9 

805.0 

812.2 

819.3 

19 

826.5 

833.7 

841.0 

848.2 

855.5 

862  9 

870.2 

877.6 

885.1 

892.5 

20 

900.0 

907.5 

915.1 

922.6 

'930.2 

937.9 

945.5 

953.2 

961.0 

968.7 

21 

976.5 

984.3 

902.2 

1000.0 

1007.9 

1015.9 

1023.8 

1031.8 

1039.9 

1047.9 

23 

1056.0 

1064.1 

1072.3 

1080.4 

1088.6 

1096.9 

1105.1 

1113.4 

1121.8 

1130  1 

23 

1138.5 

1146.9 

1155.4 

1163.8 

1172.3 

1180.9 

1189.4 

1178.0 

1206.7 

1215.3 

24 

1224.0 

1232.7 

1241.5 

1250.2 

1259.0 

1267.9 

1276.7 

1285.6 

1294.6 

1303.5 

25 

1312.5 

1321.5 

1330.6 

1339.6 

1348.7 

1357.9 

1367.0 

1376.2 

1385  5 

1394.7 

26 

1404.0 

1413.3 

1422.7 

1432.0 

1441.4 

1450.9 

1460.3 

1469.8 

1479.4 

1488.9 

27 

1498.5 

1508.1 

1517.8 

1527.4 

1537.1 

1546.9 

1556.6 

1566.4 

1576.3 

1580.1 

28 

1596.0 

1605.9 

1615.9 

1625.8 

1635.8 

1645.9 

1655.9 

1606.0 

1676.2 

1686.3 

29      1696.5 

1706.7 

1717.0 

1727.2 

1737.5 

1747.9 

1758.2 

1708.6 

1779.1 

ITS!).  5 

372 


TABLE  XVII.—  AREAS  OF   LEVEL   SECTIONS 

Base,  26  =  28  feet.     Side  slopes  1}  to  1. 


C.  H. 

.0 

.1 

.2 

.8 

.4 

.6 

.6 

.7 

.8 

.9 

0 

0.0 

2.8 

5.7 

8.5 

11.4 

14.4 

17.3 

20. 

3 

23.4 

26.4 

1 

29.5 

32.6 

35.8 

38.9 

42.1 

45.4 

48.6 

51. 

9 

55.3 

58.6 

2 

62.0 

65.4 

68.9 

72.3 

75.8 

79.4 

82.9 

86. 

5 

90.2 

93.8 

3 

97.5 

101.2 

105.0 

108.7 

112.5 

116.4 

120.2 

124. 

1 

128.1 

132.0, 

4 

136.0 

140.0 

144.1 

148.1 

152.2 

156.4 

160.5 

164. 

•J* 

169  0 

173.2 

5 

177.5 

181.8 

186.2 

190.5 

194.9 

199.4 

203.8 

208. 

3 

212.9 

217.4 

6 

222.0 

226.6 

231.3 

235.9 

240.6 

245.4 

250.1 

254. 

9 

259.8 

264  .  6 

7 

269.5 

274.4 

279.4 

284.3 

289.3 

294.4 

299.4 

304. 

5 

309.7 

314.8 

8 

320.0 

325.2 

330.5 

335.7 

341.0 

346  4 

351.7 

357. 

1 

362  6 

368.0 

9 

373.5 

379.0 

384.6 

390.1 

395.7 

401.4 

407.0 

412. 

418.5 

424.2 

10 

430.0 

435.8 

441.7 

447.5 

453.4 

459.4 

465.3 

471. 

3 

477.4 

483.4- 

11 

489.5 

495.6 

501.8 

507.9 

514.1 

520.4 

526.6 

532. 

9 

539.3 

545.6 

12 

552.0 

558.4 

564.9 

571.3 

577.8 

584.4 

590.9 

597. 

5 

604.2 

610.8 

13 

617.5 

624.2 

631.0 

637.7 

644.5 

651.4 

658.2 

665. 

1 

672.1 

679.0 

14 

686.0 

693.0 

700.1 

707.1 

714.2 

721.4 

728.5 

735. 

743.0 

750.2 

15 

757.5 

764.8 

772.2 

779.5 

786.9 

794.4 

801.8 

809. 

3 

816.9 

824.4 

16 

832.0 

839.6 

847.3 

854.9 

862.6 

870.4 

878.1 

885. 

9 

893.8 

901.6 

17 

909.5 

917.4 

925.4 

933.3 

941.3 

949.4 

957.4 

965. 

5 

973.7 

981.8 

18 

990.0 

998.2 

1006.5 

1014.7 

1023.0 

1031.4 

1039.7 

1048. 

1 

1056.6 

1065.0 

19 

1073.5 

1082.0 

1090.6 

1099.1 

1107.7 

1116.4 

1125.0 

1133. 

7 

1142.5 

1151.2 

20 

1160.0 

1168.8 

1177.7 

1186.5 

1195.4 

1204.4 

1213.3 

1222. 

3 

1231.4 

1240.4 

21 

1249.5 

1258.6 

1267.8 

1276.9 

1286.1 

1295.4 

1304.6 

1313 

9 

1323.3 

1332.6 

22 

1342.0 

1351.4 

1360.9 

1370.3 

1379.8 

1389.4 

1398.9 

1408. 

5 

1418.2 

1427.8 

23 

1437.5 

1447.2 

1457.0 

1466.7 

1476.5 

1486.4 

1496.2 

1506. 

1 

1516.1 

1526.0 

24 

1536.0 

1546.0 

1556.1 

1566.1 

1576.2 

1586.4 

1596.5 

1606 

7 

1617.0 

1627.2 

25 

1637.5 

1647.8 

1658.2 

1668.5 

1678.9 

1689.4 

1699.8 

1710. 

3 

1720.9 

1731  .4 

26 

1742.0 

1752.6 

1763.3 

1773.9 

1784.6 

1795.4 

1806.1 

1816. 

9 

1827.8 

1838.6 

27 

1849.5 

1860.4 

1871.4 

1882  3 

1893.3 

1904.4 

1915.4 

1926. 

5 

1937.7 

1948.8 

28 

1960.0 

1971.2 

1982.5 

1993  7 

2005.0 

2016.4 

2027.7 

2039. 

1 

2050.6 

2062.0 

29 

2073.5 

2085.0 

2096.6 

2108.1 

2119.7 

2131.4 

2143.0 

2154. 

7 

2166.5 

2178.2 

Base,  2ft  =  18  feet.     Side  slopes  1  to  1. 


C.  H. 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8 

.9 

0 

0.0 

1.8 

3.6 

5.5 

7.4 

9.3 

11.2 

13.1 

15.0 

17.0 

1 

19.0 

21.0 

23.0 

25.1 

27.2 

29.3 

31.4 

33.5 

35.6 

37.8 

2 

40.0 

42.2 

44.4 

46.7 

49.0 

51.3 

53.6 

55.9 

58.2 

60.6 

3 

63.0 

65.4 

67.8 

70.3 

72.8 

75.3 

77.8 

80.3 

82.8 

85.4 

4 

88.0 

90.6 

93.2 

95.9 

98.6 

101.3 

104.0 

106.7 

109.4 

112.2 

5 

115.0 

117.8 

120.6 

123.5 

126.4 

129.3 

132.2 

135.1 

138.0 

141.0 

6 

144.0 

147.0 

150.0 

153.1 

156.2 

159.3 

162.4 

165.5 

168.6 

171.8 

7 

175.0 

178.2 

181.4 

184.7 

188.0 

191.3 

194.6 

197.9 

201.2 

204.6 

8 

208.0 

211.4 

214.8 

218.3 

221  8 

225.3 

228.8 

232.3 

235.8 

239.4 

9 

243.0 

246.6 

250.2 

253.9 

257.6 

261.3 

265.0 

268.7 

272.4 

276.2 

10 

280.0 

283.8 

287.6 

291.5 

295.4 

299.3 

303.2 

307.1 

311.0 

315.0 

11 

319.0 

323.0 

327.0 

331.1 

335.2 

339.3 

343.4 

347.5 

351.6 

355.8 

12 

360.0 

364.2 

368.4 

372.7 

377.0 

381.3 

385.6 

389.9 

394.2 

398.6 

13 

403.0 

407.4 

411.8 

416.3 

420.8 

425.3 

429.8 

434.3 

438.8 

443.4 

14 

448.0 

452.6 

457.2 

461.9 

466.6 

471.3 

476.0 

480.7 

485.4 

490.2 

15 

495.0 

499.8 

504.6 

509.5 

514.4 

519.3 

524.2 

529.1 

534.0 

539.0 

16 

544.0 

549.0 

554.0 

559.1 

564.2 

569.3 

574.4 

579.5 

584.6 

589.8 

17 

595.0 

600.2 

605.4 

610.7 

616.0 

621.3 

626.6 

631.9 

637.2 

642.6 

18 

648.0 

653  4 

658.8 

664.3 

669.8 

675.3 

680.8 

686.3 

691.8 

697.4 

19 

703.0 

708.6 

714.2 

719.9 

725.6 

731  3 

737.0 

742.7 

748.4 

754.2 

20 

760.0 

765.8 

771.6 

777.5 

783.4 

789.3 

795.2 

801.1 

807.0 

813.0 

21 

819.0 

825.0 

831.0 

837.1 

843.2 

849.3 

855.4 

861.5 

867.6 

873.8 

22 

880.0 

886.2 

892.4 

898.7 

905.0 

911.3 

917.6 

923.9 

930.2 

936.6 

23 

943.0 

949.4 

955.8 

962.3 

968.8 

975.3 

981.8 

988.3 

994.8 

1001.4 

24 

1008.0 

1014.6 

1021.2 

1027.9 

1034.6 

1041.3 

1048.0 

1054.7 

1061.4 

1068.2 

25 

1075  0 

1081.8 

1088.6 

1095.5 

1102.4 

1109.3 

1116.2 

1123.1 

1130.0 

1137.0 

26 

1144  0 

1151.0 

1158.0 

1165.1 

1172.2 

1179.3 

1186.4 

1193.5 

1200.6 

1207.8 

27 

1215.0 

1°22  2 

1229.4 

1236.7 

1244.0 

1251.3 

1256.6 

1265.9 

1273.2 

1280.6 

28 

1288.0 

1295'  4 

1302.8 

1310.3 

1317.8 

1325.3 

1332.8 

1340.3 

1347.8 

1355.4 

29 

1363.0 

1370.6 

1378.2 

1385.9 

1393.6 

1401.3 

1409.0 

1416.7 

1424  4 

1432.2 

TABLE  XVII. -ARE AS  OP  LEVEL  SECTIONS. 
Base,  26  =  20  feet.    Side  slopes  1  to  1. 


C.  H. 


.0          .1          .2          .3          .4          .5          .6          .7          .8 


7 

8 
9 

10 
11 
12 
13 
14 
15 
10 
17 
IS 
19 
20 
21 
22 
23 
24 
25 
27 
2(5 
28 
29 


0.0  2.0  4.0        6.1  8.2  10.3  12.4  14.5  16.6      18.8 

21.0  23.2  25.4      27.7  30.0  32.3  34.6  36.9  39.2      41.6 

44.0  46.4  48.8      51.3  53.8  56.3  58.8  61.3  63.8      66.4 

69.0  71.6  74.2      76.9  79.6  82.3  85.0  87.7  90.4      93.2 

96.0  98.8  101.6     104.5  107.4  110.3  113.2  116.1  119.0    122.0 

125.0  128.0  131.0     134.1  137.2  140.3  143.4  146.5  J49.6    152.8 

156.0  159.2  162.4     165.7  169.0  172.3  175.6  178.9  182.2    185.6 

189.0  192.4  195.8     199.3  202.8  206.3  209.8  213.3  216.8    220.4 

224.0  227.6  231.2     234.9  238.6  242.3  246.0  249.7  253.4    257.2 

261.0  264.8  268.6     272.5  276.4  280.3  284.2  288.1  292.0    296.0 

300  0  304. C  308.0     312.1  316.2  320.3  324.4  328.5  332.6    336.8 

341.0  345.2  349.4     353.7  358.0  362.3  366.6  370.9  375.2    379.6 

384.0  388.4  392.8     397.3  401.8  406.3  410.8  415.3  419.8    424.4 

4^9.0  433.6  438.2     442.9  447.6  452.3  457.0  461.7  466.4    471.2 

476.0  480.8  485.6     490.5  495.4  500.3  505.2  510.1  515.0    520.0 

525.0  530.0  535.0     540.1  545.2  550.3  555.4  560.5  565.6    570.8 

576.0  581.2  586.4     591.7  597.0  602.3  607.6  612.9  618.2    623.6 

629.0  634.4  639.8     645.3  650.8  656.3  661.8  667.3  672.8    678.4 

684.0  689.6  695.2     700.9  706.6  712.3  718.0  723.7  729.4    735.2 

741.0  746.8  752.6     758.5  764.4  770.3  776.2  782.1  788.0    794.0 

800.0  806.0  812.0    818.1  824.2  830.3  836.4  842.5  848.6    854.8 

861.0  867.2  873.4    879.7  886.0  892.3  898.6  904.9  911.2    917.6 

924.0  930.4  936.8    943.3  949.8  956.3  962.8  969.3  975.8    982.4 

989.0  995.6  1002.2  1008.9  1015.6  1022.3  1029.0  1035.7  1042.4  1049.2 

1056.0  1062.8  1069.6  1076.5  1083.4  1090.3  1097.2  1104.1  1111.0  1118.0 

1125.0  1132.0  1139.0  1146.1  1153.2  1160.3  1167.4  1174.5  1181.6  1188.8 

1196.0  1203.2  1210.4  1217.7  1225.0  1232.3  1239.6  1246.9  1254.2  1261.6 

1269.0  1276.4  1283.8  1291.3  1298.8  1306.3  1313.8  1321.3  1328.8  1336.4 

1344.0  1351.6  1359.2  1366.9  1374.6  1382.3  1390.0  1397.7  1405.4  1413.2 

1421.0  1428.8  1436.6  1444.5  1452.4  1460.3  1468.2  1476  1  1484.0  1492.0 


Base,  26  =  30  feet.    Side  slopes  1  to  1. 


C.  H. 

.0 

.1          .2          .3          .4          .6          .6         ,7         .8         .9 

0 

0.0 

3.0         6.0        9.1       12.2       15.3      18.4      21.5      24.6      27.8 

1 

31.0 

34.2       37.4      40.7      44.0      47.3      50.6      53.9      57.2      60.6 

2 

64.0 

67.4      70.8      74.3      77.8      81.3      84.8      88.3      91.8      95.4 

3 

99.0 

102.6     106.2     109.9     113.6     117.3    121.0    124.7    128.4    132.2 

4 

136.0 

139.8     143.6     147.5     151.4     155.3     159.2    163.1     167.0    171.0 

5 

175.0 

179.0     183.0     187.1     191.2    195.3    199.4    203.5    207.6    211.8 

6 

216.0 

220.2     224.4     228.7     233.0     237.3    241.6    245.9    250.2    254.6 

7 

259.0 

263.4     267.8     272.3    276.8     281.3    285.8    290.3    294.8    299.4 

8 

304.0 

308.6     313.2     317.9     322.6     327.3     332.0    336.7    341.4    346.2 

9 

351.0 

355.8     360.6     365.5    370.4     375.3    380.2    385.1     390.0    395.0 

10 

400.0 

405.0     410.0    415.1     420.2     425.3    430.4    435.5    440.6    445.8 

11 

451  .0 

456.2     461.4     466.7    472.0     477.3    482.6    487.9    493.2    498.6 

12 

504.0 

509.4     514.8     520.3     525.8     531.3    536.8    542.3    547.8    553.4 

13 

559.0 

564.6     570.2    575.9     581.6     587.3    593.0    598.7    604.4    610.2 

14 

616.0 

621.8     627.6     633.5     639.4     645.3    651.2    657.1     663.0    669.0 

15 

675.0 

681.0     687.0    693.1     699.2    705.3    711.4    717.5    723.6    729.8 

16 

736.0 

742.2     748.4     754.7     761.0    767.3    773.6    779.9    786.2    792.6 

17 

799.0 

805.4     811.8     818.3     824.8     831.3    837.8    844.3    850.8    857.4 

18 

864.0 

870.6     877.2    883.9    890.6     897.3    904.0    910.7    917.4    924.2 

19 

931.0 

937.8     944.6     951.5    958.4     965.3    972.2    979.1    986.0    993.0 

20 

1000.0 

1007.0   1014.0  10-21.1   1028.2  1035.3  1042.4  1049.5  1056.6  1063.8 

21 

1071.0 

1078.2   1085.4   1092.7  1100.0  1107.3  1114.6  1121.9  1129.2  1136.6 

22 

1144.0 

1151.4   1158.8  1166.3  1173.8  1181.3  1188.8  1196.3  1203.8  1211.4 

23 

1-J19.0 

1226.6   1234.2  1241.9  1249.6  1257.3  1265.0  1272.7  1280.4  1288.2 

24 

1296.0 

1303.8   1311.6  1319.5  1327.4  1335.3  1343.2  1351.1  1359.0  1367.0 

25 

1375.0 

1383.0   1391.0  1399.1   1407.2  1415.3  1423.4  1431.5  1439.6  1447.8 

26 

1456.0 

1464.2   1472.4  1480.7  1489.0  1497.3  1505.6  1513.9  1522.2  1530.6 

27 

1539.0 

1547.4   1555.8  1564.3  1572.8  1581.3  1589.8  1598.3  1(106.  8  1(515.  -1 

28 

1624.0 

1632.6   1641.2  1649.9  1658.6  1667.3  1676.0  1684.7  1693.4  1702.2 

29 

1711.0 

1719.8   1728.6  1737.5  1746.4  1755.3  1764.2  1773.1  1782.0  1791.0 

TABLE   XVII.— AREAS  OF  LEVEL   SECTIONS. 
Base,  2b  =  16  feet.    Side  slopes  £  to  1. 


C.  H. 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.1 

.8 

.« 

0 

0.0 

1.6 

3.2 

4.8 

6.4 

8.1 

9.7 

11.3 

13.0 

14 

.6 

1 

16.3 

17.9 

19.6 

21.2 

22.9 

24.6 

26.2 

27.9 

29.6 

31 

.3 

2 

33.0 

34.7 

36.4 

38.1 

39.8 

41.6 

43.3 

45.0 

46.8 

48 

.5 

3 

50.3 

52.0 

53.8 

55.5 

57.3 

59.1 

60.8 

62.6 

64.4 

0(5 

.2 

4 

68.0 

69.8 

71.6 

73.4 

75.2 

77.1 

78.9 

80.7 

82.6 

84 

.4 

6 

8(5.3 

88.1 

90.0 

91.8 

93.7 

95.6 

97.4 

99.3 

101.2 

103 

.1 

0 

105.0 

106.9 

108.8 

110.7 

112.6 

114.6 

116.5 

118.4 

120.1 

122 

.3 

7 

124.3 

126.2 

128.2 

130.1 

132.1 

134.1 

136.0 

138.0 

140.0 

142 

.0 

8 

144.0 

146.0 

148.0 

150.0 

152.0 

154.1 

156.1 

158.1 

160.2 

162 

.2 

9 

164.3 

166.3 

168.4 

170.4 

172.5 

174.6 

176.6 

178.7 

180.8 

188 

.9 

10 

185.0 

187.1 

189.2 

191.3 

193.4 

195.6 

197.7 

199.8 

202.0 

204 

.1 

11 

206.3 

208.4 

210.6 

212.7 

214.9 

217.1 

219.2 

221.4 

223.6 

JW5 

.8 

18 

228.0 

230.2 

232.4 

234.6 

236.8 

239.1 

241.3 

243.5 

245.8 

348 

.0 

13 

250.3 

252.5 

254.8 

257.0 

259.3 

261.6 

263.8 

266.1 

268.4 

270 

7 

14 

273.0 

275.3 

277.6 

279.9 

282.2 

284.6 

286.9 

289.2 

291.6 

293 

!9 

15 

296.3 

298.6 

301.0 

303.3 

305.7 

308.1 

310.4 

312.8 

315.2 

317 

.6 

16 

320.0 

322.4 

324.8 

327.2 

329.6 

332.1 

334.5 

336.9 

&S9.4 

341 

.8 

17 

344.3 

346.7 

349.2 

351.6 

354.1 

356.6 

359.0 

361.5 

364.0 

366 

.5 

18 

369.0 

371.5 

374.0 

876.5 

379.0 

381.6 

384.1 

386.6 

389.2 

391 

7 

19 

394.3 

396.8 

399.4 

401.9 

404.5 

407.1 

409.6 

412.2 

414.8 

417 

'A 

20 

420.0 

422.6 

425.2 

427.8 

430.4 

433.1 

435.7 

438.3 

441.0 

443 

.6 

2  1 

446.3 

448.9 

451.6 

454.2 

456.9 

459.6 

462.2 

464.9 

467.6 

470 

.3 

22 

473.0 

475.7 

478.4 

481.1 

483.8 

486.6 

489.3 

492.0 

494.8 

497 

.5 

23 

500.3 

503.0 

505.8 

508.5 

511.3 

514.1 

516.8 

519.6 

522.4 

r>-,>r 

.2 

24 

528.0 

530.8 

533.6 

536.4 

539.2 

542.1 

544.9 

547.7 

550.6 

553 

.4 

25 

556.3 

559.1 

562.0 

564.8 

567.7 

570.6 

573.4 

576.3 

579.2 

582 

.1 

26 

585.0 

587.9 

590.8 

593.7 

596.6 

599.6 

602.5 

605.4 

608.4 

611 

.3 

27 

614.3 

617.2 

620.2 

623.1 

626.1 

629.1 

632.0 

635.0 

638.0 

641.0 

28 

644.0 

647.0 

650.0 

653.0 

656.0 

659.1 

662.1 

665.1 

668.2 

671 

.2 

29 

674.3 

677.3 

680.4 

683.4 

686.5 

689.6 

692.6 

695.7 

698.8 

701 

.9 

Base,  2ft  =  18  feet.    Side  slopes  £  to  1. 


C.  H. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

1.8 

3.6 

5.4 

7.2 

9.1 

10.9 

12.7 

14.6 

16.4 

1 

18.3 

20.1 

2->.0 

23.8 

25.7 

27.6 

29.4 

31.3 

33.2 

35.1 

2 

37.0 

38.9 

40.8 

42.7 

44.6 

46.6 

48.5 

50.4 

52.4 

54.3 

3 

55.  a 

58.2 

60.2 

62.1 

64.1 

66.1 

68.0 

70.0 

.0 

74.0 

4 

76.0 

78.0 

80.0 

82.0 

84.0 

86.1 

88.1 

90.1 

.2 

94.2 

r> 

96.3 

98.3 

100.4 

102.4 

104.5 

106.6 

108.6 

110.7 

112.8 

114.9 

6 

117.0 

119.1 

121.2 

123.3 

125.4 

127.6 

129.7 

131.8 

134.0 

136.1 

7 

138.3 

140.4 

142.6 

144.7 

146.9 

149.1 

151.2 

153.4 

155.6 

157.8 

8 

IfiO.O 

162.2 

164.4 

166.6 

168.8 

171.1 

173.3 

175.5 

177.8 

180.0 

9 

182.3 

184.5 

186.8 

189.0 

191.3 

193.6 

195.8 

198.1 

200.4 

202.7 

10 

205.0 

207.3 

209.6 

211.9 

214.2 

216.6 

218.9 

221.2 

223.6 

225.9 

11 

228.3 

230.6 

233.0 

235.3 

237.7 

240.1 

242.4 

244.8 

247.2 

249.6 

12 

252.0 

254.4 

256.8 

259.2 

261.6 

264.1 

266.5 

268.9 

271.4 

273.8 

13 

276.3 

278.7 

281.2 

283.6 

286.1 

288.6 

291.0 

293.5 

296.0 

298.5 

14 

301.0 

303.5 

306.0 

308.5 

311.0 

313.6 

316.1 

318.6 

321.2 

323.7 

1  •") 

326.3 

328.8 

331.4 

333.9 

336.5 

339.1 

341.6 

344.2 

346  8 

349.4 

16 

352  0 

354.6 

357.2 

359.8 

362.4 

365.1 

367.7 

370.3 

373.0 

375.6 

1  7 

378.3 

380.9 

383.6 

386.2 

388.9 

391.6 

394.2 

396.9 

3y9.6 

402.3 

18 

405.0 

407.7 

410.4 

413.1 

415.8 

418.6 

421.3 

424  0 

426.8 

429.5 

19 

432.3 

435.0 

437.8 

440.5 

443.3 

44(5.1 

448.8 

451.6 

454.4 

457.2 

20 

460.0 

462.8 

465.6 

468.4 

471.2 

474.1 

476.9 

479.7 

482.6 

485.4 

21 

488.3 

491.1 

494.0 

496.8 

499.7 

502.6 

505.4 

508.3 

511.2 

514.1 

22 

517.0 

519.9 

522.8 

525.7 

528.6 

531.6 

534.5 

537.4 

540.4 

543.3 

23 

546.3 

549.2 

552.2 

555.1 

558.1 

561.1 

5(54.0 

567.0 

570.0 

573.0 

24 

576.0 

579.0 

582.0 

585.0 

588.0 

591  .  1 

594.1 

597.1 

600.2 

603.2 

25 

606.3 

609.3 

612.4 

615.4 

618.5 

621.6 

624.6 

627.7 

630.8 

633.9 

26 

637.0 

640.1 

643.2 

646.3 

649.4 

652.6 

655.7 

658.8 

662.0 

665.1 

27 

668.3 

671.4 

674.6 

677.7 

680.9 

684.1 

687.2 

690.4 

693.6 

(596.8 

28 

700.0 

703.2 

706.4 

709.6 

712.8 

716.1 

719.3 

722.5 

725.8 

729.0 

29 

732.3 

735.5 

738.8 

742.0 

745.3 

748.6 

751.8 

755.1 

758.4 

761.7 

XVIII.— AREA  CORRECTIONS  FOR  THREE-LEVEL  GROUND.    375 
Correction  =  (hm  -  7i0)2s.    (See  331.) 

SIDE   SLOPES    U   TO    1. 


hm-h0 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0. 

0 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1 

1. 

5 

1.8 

2.2 

2.5 

2.9 

3.4 

3.8 

4.3 

4.9 

5.4 

2 

6. 

0 

6.6 

7.3 

7.9 

8.6 

9.4 

10.1 

10.9 

11.8 

12.6 

3 

13. 

5 

14.4 

15.4 

16.3 

17.3 

18.4 

19.4 

20.5 

21.7 

22.8 

4 

24. 

0 

25.2 

26.5 

27.7 

29.0 

30.4 

31.7 

33.1 

34.6 

36.0 

5 

37. 

5 

89.0 

40.6 

42.1 

43.7 

45.4 

47.0 

48.7 

50.5 

52.2 

6 

54. 

0 

55.8 

57.7 

59.5 

61.4 

63.4 

65.3 

67.3 

69.4 

71.4 

7 

73. 

5 

75.6 

77  .8 

79.9 

82.1 

84.4 

86.6 

88.9 

91.3 

93.6 

8 

96. 

0 

98.4 

100.9 

103.3 

105.8 

108.4 

110.9 

113.5 

116.2 

118.8 

9 

121. 

5 

124.2 

127.0 

129.7 

132.5 

135.4 

138.2 

141.1 

144.1 

147.0 

10 

150. 

0 

153.0 

156.1 

159.1 

162.2 

165.4 

168.5 

171.7 

175.0 

178.2 

11 

181. 

5 

184.8 

188.2 

191.5 

194.9 

198.4 

201.8 

205.3 

208.9 

212.4 

SIDE  SLOPES   1   TO   1. 


/im-fco 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8 

.9 

0 

.    o.o 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

1 

1.0 

1.2 

1.4 

1.7 

2.0 

2.3 

2.6 

2.9 

3  2 

3.6 

2 

4.0 

4.4 

4.8 

5.3 

5.8 

6.3 

6.8 

7.3 

7!s 

8.4 

3 

9.0 

9.6 

10.2 

10.9 

11.6 

12.3 

13.0 

13.7 

14.4 

15.2 

4 

16.0 

16.8 

17.6 

18.5 

19.4 

20.3 

21.2 

22.1 

23.0 

24.0 

5 

25.0 

26.0 

27.0 

28.1 

29.2 

30.3 

31.4 

32.5 

33.6 

34.8 

G 

36.0 

37.2 

38.4 

39.7 

41.0 

4','.  3 

43.6 

44.9 

46.2 

47.6 

7 

49.0 

50.4 

51.8 

53.3 

54.8 

56.3 

57.8 

59.3 

60.8 

62.4 

8 

64.0 

65.6 

67.2 

68.9 

70.6 

72.3 

74.0 

75.7 

77.4 

79.2 

g 

81  0 

82  8 

84.6 

86.5 

88.4 

90.3 

92  2 

94.1 

96  0 

98  0 

10 

100.0 

102.0 

104.0 

106.1 

108.2 

110.3 

112.4 

114.5 

116.6 

118.8 

11 

121.0 

123.2 

125.4 

127.7 

130.0 

132.3 

134.6 

136.9 

139.2 

141.6 

SIDE   SLOPES   J  TO   1. 


hm-h0 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

1 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

1.3 

1.4 

1.6 

1.8 

2 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.4 

3.6 

3.9 

4.2 

3 

4.5 

4.8 

5.1 

5.4 

5.8 

6.1 

6.5 

6.8 

7.2 

T.6 

4 

8.0 

8.4 

8.8 

9.2 

9.7 

10.1 

10.6 

11.0 

11.5 

12.0 

5 

12.5 

13.0 

13.5 

14.0 

14  6 

15.1 

15.7 

16.2 

16.8 

17.4 

6 

18.0 

18.6 

19.2 

19.8 

20.5 

21.1 

21.8 

22.4 

23.1 

23.8 

7 

24.5 

25.2 

25.9 

26.6 

27.4 

28.1 

28.9 

29.6 

30.4 

31.2 

8 

32.0 

32.8 

33.6 

34.4 

35.3 

36.1 

37.0 

37.8 

38.7 

39.6 

9 

40.5 

41.4 

42.3 

43.2 

44.2 

45.1 

46.1 

47.0 

48.0 

49.0 

10 

50.0 

51.0 

52.0 

53.0 

54.1 

55.1 

56.2 

57.2 

58.3 

59.4 

11 

60.5 

61.6 

62.7 

63.8 

65.0 

66.1 

67.3 

68.4 

69.6 

70.8 

IDE   SLOPES 

i  TO  1. 

hm-h0 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

1 

0.3 

0.3, 

0.4 

0.4 

0.5 

0.6 

0.6 

0.7 

0.8 

0.9 

2 

1.0 

1.1 

1.2 

1.3 

1.4 

1.6 

1.7 

.    1.8 

2.0 

2.1 

3 

23 

2.4 

2.6 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.8 

4 

4.0 

4.2 

4.4 

4.6 

4.8 

5.1 

5.3 

5.5 

5.8 

6.0 

i 

6  3 

6.5 

6.8 

7.0 

7.3 

7.6 

7.8 

8.1 

8.4 

8.7 

0 

9.0 

9  3 

9.6 

9.9 

10.2 

10.6 

10.9 

11.2 

11.6 

11.9 

7 

12.3 

12.6 

13.0 

13.3 

13.7 

14.1 

14.4 

14.8 

15.2 

15.6 

8 

16.0 

16.4 

16.8 

17.2 

17.6 

18.1 

18.5 

18.9 

19.4 

19.8 

9 

20.3 

20.7 

21.2 

21.6 

22.1 

22.6 

23.0 

23.5 

24.0 

24.5 

10 

25.0 

25.5 

26.0 

26.5 

27.0 

27.6 

28.1 

28.6 

29.2 

29.7 

11 

30.3 

30.8 

21.4 

31.9 

32.5 

33.1 

33.6 

34.2 

34.8 

35.4 

376  xix.— CUBIC  YARDS  PER  100  FEET.   SLOPES  ± :  i. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

45 

53 

60 

68 

82 

90 

97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

370 

400 

430 

5 

245 

282 

319 

356 

431 

468 

505 

542 

6 

300 

344 

389 

433 

522 

567 

611 

656 

7 

356 

408 

460 

612 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

608 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

907 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

1090 

1171 

1253 

12 

667 

756 

844 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

1023 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

873 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

1184 

1304 

1541 

1659 

1778 

1896 

17 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

2164 

2305 

20 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

22 

1426 

1589 

1752 

1915 

2241 

2404 

2567 

S730 

23 

1512 

1682 

1&53 

2023 

2364 

2534 

2705 

2875 

24 

1600 

1778 

1955 

2133 

2489 

2667 

2844 

3022 

25 

1690 

1875 

2060 

2245 

2616 

.2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

3322 

27 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

28 

1970 

2178 

2384 

2593 

3007 

3215 

3422 

3630 

29 

2068 

2282 

2496 

2712 

3142 

.3356 

3571 

3786 

30 

2167 

2389 

2610 

2833 

3278 

3500 

3722 

3944 

31 

2268 

2497 

2726 

2956 

3416 

3645 

3875 

4105 

32 

2370 

2607 

2844 

3081 

3556 

3793 

4030 

4267 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

2833 

3085 

3337 

3841 

4093 

4344 

4596 

35 

2690 

2949 

3208 

3468 

3986 

-4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

4133 

4400 

4667 

4933 

37 

2912 

3186 

3460 

3734 

4282 

4556 

4831 

5105 

38 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

5278 

39 

3142 

3431 

3719 

4008 

4586 

4875 

5164 

5453 

40 

3259 

3556 

3852 

4148 

4741 

5037 

5333 

5630 

41 

3379 

3682 

3986 

4290 

4897 

5201 

5505 

5808 

42 

3500 

3811 

4122 

4433 

5056 

5367 

5678 

5SS9 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875 

6208 

6542 

46 

4004 

4344 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

4830 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4622 

4978 

5333 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490 

6216 

6579 

6942 

7305 

50 

4537 

4907 

5278 

5648 

6389 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

597'0 

6741 

7126 

7511 

7896 

53 

4956 

•5349 

5741 

6134 

6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300 

7100 

7500 

7900 

8300 

55 

5245 

5653 

6060 

6468 

7282 

7690 

8097 

8505 

56 

5393 

5807 

6222 

6637 

7467 

7881 

8296 

8711 

57 

5542 

5964 

6386 

6808 

7653 

8075 

8497 

8919 

58 

5693 

6122 

6552 

6981 

7841 

3270 

8700 

9130 

59 

5845 

6282 

6719 

7156 

8031 

8468 

8905 

9342 

60 

6000 

6444 

6889 

7333 

8222 

8667 

9111 

9556 

XIX.— CUBIC  YARDS  PER  100  FEET.     SLOPES  \  :  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

46 

54 

61 

69 

83 

91 

98 

106 

2 

96 

111 

126 

141 

170 

185 

200 

215 

3 

150 

172 

194 

217 

261 

283 

306 

328 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

454 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

7 

402 

454 

506 

557 

661 

713 

765 

817 

8 

474 

533 

593 

652 

770 

830 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202 

1283 

1365 

12 

800 

889 

978 

1067 

1244 

1333 

1422 

1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

1001 

14  ' 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1366 

1417 

1639 

1750 

1801 

1972 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

17 

1291 

1417 

1543 

1669 

1920 

2046 

2172 

2298 

18 

1400 

1533 

1667 

1800 

2067 

2200 

2333 

2467 

19 

1513 

1654 

1794 

1935 

2217 

2357 

2498 

2639 

20 

1630 

1778 

1926 

2074 

2370 

2519 

2667 

2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

3365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

25 

2269 

2454 

2639 

2824 

3194 

3380 

3565 

3750 

26 

2407 

2600 

2793 

2985 

3370 

35  C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

3550 

3750 

3950 

4151 

28 

2696 

2904 

3111 

3319 

3733 

3941 

4148 

4356 

29 

2846 

3061 

3276 

3491 

3920 

4135 

4350 

4565 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

31 

3157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

3972 

4217 

4706 

4950 

5194 

5439 

34 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

5667 

&5 

3824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133 

37 

4180 

4454 

4728 

5002 

5550 

5824 

6098 

6372 

38 

4363 

4644" 

4926 

5207 

5770 

6052 

6333 

6015 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6801 

40 

4741 

5037 

5333 

5630 

6222 

6519 

6815 

7111 

41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

42 

5133 

5444 

5756 

6067 

6089 

7000 

7311 

7622 

43 

5335 

5654 

5972 

6291 

6928 

7246 

7505 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7822 

8148 

45 

5750 

6083 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

6644 

6985 

7667 

8007 

8348 

8689 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8017 

8965 

48 
49 

6400 
6624 

6756 

6987 

7111 
7350 

£67 
7*  Jo 

S 

m 

9044 

9528 

50 

6852 

7222 

7593 

7903 

87C4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52 

7319 

7704 

8089 

8474 

9244 

9630 

10015 

10-100 

53 

7557 

7950 

8343 

87:35 

9520 

9913 

10306 

10098 

54 

7800 

8200 

8600 

9000 

9800 

10200 

10600 

11000 

55 

8046 

8454 

8861 

9269 

10083 

10491 

10898 

11300 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11015 

57 

8550 

8972 

9394 

9817 

10601 

11083 

11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385 

11815 

12244 

59 

9009 

9506 

9943 

10380 

11254 

11091 

12128 

12565 

60 

9333 

9778 

10222 

10667 

1155G 

12000 

12444 

12889 

378  XIX.-CtTBlC  YARDS  PER  100  FEET.     SLOPES  1  :  1. 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

20 

28 

30 

32 

j 

48 

56 

G3 

70 

78 

107 

115 

122 

2 

104 

119 

133 

148 

163 

222 

237 

252 

3 

167 

189 

211 

233 

256 

344 

367 

389 

4 

'237 

267 

296 

326 

356 

474 

504 

533 

5 

315 

352 

389 

426 

463 

611 

648 

685 

G 

400 

444 

489 

533 

578 

756 

800 

844 

7 

493 

544 

596 

648 

700 

907 

959 

1011 

8 

593 

652 

711 

770 

830 

1067 

1126 

1185 

9 

700 

767 

833 

900 

9G7 

1233 

1300 

1367 

10 

815 

889 

963 

1037 

1111 

1407 

1481 

1556 

11 

937 

1019 

1100 

1181 

1263 

1589 

1670 

1752 

1-2 

1067 

1156 

1244 

1S33 

1422 

1778 

1867 

1956 

13 

1204 

1300 

1396 

1493 

1589 

1974 

2070 

2167 

14 

1348 

1452 

1556 

1659 

1763 

2178 

2281 

2385 

15 

1500 

1611 

1722 

1833 

1944 

2389 

2500 

2611 

16 

1659 

1778 

1896 

2015 

2133 

2607 

2726 

2844 

17 

1826 

1952 

2078 

2204 

2330 

2833 

2959 

3085 

18 

2000 

2133 

2267 

2400 

2533 

3067 

3200 

3333 

19 

2181 

2322 

2463 

2604 

2744 

3307 

3448 

8589 

20 

2370 

2519 

2G07 

2815 

2963 

3556 

3704 

3852 

21 

2567 

2722 

2878 

3033 

3189 

3811 

3967 

4122 

22 

2770 

2933 

3096 

3259 

3422 

4074 

4237 

4144 

23 

2981 

3152 

3322 

3493 

3663 

4344 

4515 

4685 

24 

3200 

3378 

3556 

3733 

3911 

4622 

4800 

4978 

25 

3426 

3611 

3796 

3981 

4167 

4907 

5093 

5278 

26 

3659 

3852 

4044 

4237 

4430 

5200 

5393 

5585 

27 

3900 

4100 

4300 

4500 

4700 

5500 

5700 

5900 

28 

4148 

4356 

4563 

4770 

4978 

5807 

6015 

6222 

29 

4404 

4619 

4833 

5048 

5263 

6122 

6337 

6552 

30 

46C7 

4889 

5111 

5333 

5556 

6444 

6667 

6889 

31 

4937 

5167 

5396 

5626 

5856 

6774 

7004 

7233 

32 

5215 

5452 

5689 

5926 

6163 

7111 

7348 

7585 

33 

5500 

5744 

5989 

6233 

6478 

7456 

7700 

7944 

34 

5793 

6044 

6296 

6548 

6800 

7807 

8059 

8311 

35 

6093 

6352 

6611 

687'0 

7130 

8167 

8426 

8685 

36 

6400 

6667 

6933 

7200 

7407 

8533 

8800 

9067 

37 

6715 

6989 

7263 

7537 

7811 

8907 

9181 

9456 

38 

7037 

7319 

7GOO 

7881 

8163 

'  9289 

9570 

9852 

39 

7367 

7656 

7944 

8233 

8522 

9678 

9967 

10256 

40 

7704 

8000 

8296 

8593 

8889 

10074 

10370 

10667 

41 

8048 

8352 

8656 

8959 

9263 

10478 

10781 

11085 

42 

8400 

8711 

9022 

9333 

9644 

10889 

11200 

11511 

43 

8759 

9078 

9396 

9715 

10033 

11307 

11626 

11914 

44 

9126 

9452 

9778 

10104 

10430 

11733 

12059 

12385 

45 

9500 

9833 

10167 

10500 

10833 

12167 

12500 

12833 

46 

9881 

10222 

10563 

10904 

11244 

12607 

12948 

13289 

47 

1027'0 

10619 

10967 

11315 

11G63 

13G56 

1:3404 

13752 

48 

10667 

110X2 

11378 

11733 

12089 

113511 

13867 

142°2 

49 

11070 

11433 

11796 

12159 

12522 

13974 

14337 

14700 

50 

11481 

11858 

12222 

12593 

12963 

14444 

14815 

15185 

51 

11900 

12278 

12656 

13033 

13411 

14922 

15300 

15678 

52 

12326 

12711 

13096 

13481 

13867 

15407 

15793 

16178 

53 

12759 

13152 

13544 

13937 

14330 

15900 

16293 

16685 

54 

13200 

13GOO 

14000 

14400 

14800 

16400 

16800 

17200 

55 

13048 

14056 

14463 

14870 

15278 

16907 

17315 

17722 

56 

14104 

14519 

14933 

15348 

157G3 

17422 

17837 

18252 

57 

14567 

14989 

15411 

15833 

16256 

17944 

183(57 

187"89 

58 

15037 

15467 

15896 

16326 

16756 

18474 

18904 

19333 

59 

15515 

15952 

16389 

16896 

17263 

19011 

19118 

1«.)SS5 

60 

16000 

16444 

16889 

17333 

17778 

19556 

20000 

20444 

xix.— CUBIC  YARDS  PER  100  FEET.   SLOPES  u :  i. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

50 

57 

65 

72 

80 

109 

117 

124 

2 

111 

126 

141 

156 

170 

230 

244 

259 

3 

ias 

206 

228 

250 

272 

361 

383 

406 

4 

267 

296 

326 

356 

385 

504 

533 

563 

5 

361 

398 

435 

472 

509 

657 

694 

731 

6 

467 

511 

556 

600 

644 

822 

867 

911 

7 

583 

635 

687 

739 

791 

998 

1050 

1102 

8 

711 

770 

830 

889 

948 

1185 

1244 

1304 

9 

850 

917 

983 

1050 

1116 

1383 

1450 

1517 

10 

1000 

1074 

1148 

1222 

1296 

1593 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1333 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

2644 

2748 

15 

1917 

2028 

2139 

2250 

2361 

2806 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

3319 

1? 

2361 

2487 

2613 

2739 

2865 

3369 

3494 

3620 

18 

2600 

2733 

2867 

3000 

3133 

3667 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

20 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

22 

3667 

3830 

3993 

4156 

4318 

4970 

5133 

5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494 

5665 

24 

4267 

4444 

4622 

4800 

4978 

5689 

5867 

6044 

23 

4583 

4769 

4954 

5139 

5324 

6065 

6250 

6435 

26 

4911 

5104 

5296 

5489 

5681 

6452 

6644 

6837 

2? 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

29 

5961 

6176 

6391 

6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

32 

7111 

7348 

7585 

7822 

8059 

9007 

9244 

9482 

33 

7517 

7761 

8006 

8250 

8494 

9472 

9717 

9962 

34 

7933 

8185 

8437 

86S9 

8941 

9948 

10200 

10452 

35 

8361 

8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800 

9067 

9333 

9600 

9867 

10933 

11200 

11467 

3? 

38 

9250 
9711 

9524 
9993 

9798 
10274 

10072 
10556 

10346 
10837 

11443 
11963 

11717 
12244 

11991 
1252B 

39 

10183 

10472 

10761 

11050 

11339 

12494 

12783 

13072 

40 

10667 

10963 

11259 

11556 

11852 

13037 

13333 

13630 

41 

11161 

11465 

11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978 

12289 

12600 

12911 

14156 

14467 

14778 

43 

12183 

12502 

12820 

13139 

13457 

14731 

15050 

15369 

44 

12711 

13037 

13363 

13689 

14015 

15319 

15644 

15970 

45 

13250 

13583 

13917 

14250 

14583 

15917 

16250 

16583 

46 

13800 

14141 

14481 

14822 

15103 

16526 

16867 

17207 

47 

14361 

14709 

15057 

15406 

15754 

17146 

17494 

17843 

48 

14933 

15289 

15644 

16000 

16356 

17778 

18183 

18489 

49 

15517 

15880 

16243 

16606 

16968 

18420 

18783 

19146 

50 

16111 

16481 

16852 

17222 

17592 

19074 

19444 

19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117 

20494 

52 

17333 

17719 

18104 

18489 

18874 

20415 

20800 

21185 

53 

179(51 

18351 

18746 

19139 

19531 

21102 

21494 

21887 

54 

18600 

19000 

19400 

19800 

20200 

21800 

22200 

22600 

55 

19250 

19657 

20065 

20472 

20880 

22509 

22917 

23324 

56 

19911 

20326 

20741 

21156 

21570 

23230 

23644 

24059 

57 

20583 

21006 

21428 

21850 

22272 

23961 

24383 

24805 

58 

21267 

21696 

22126 

2255(5 

22985 

24704 

251:33 

25563 

f    59 

21961 

22398 

22835 

23273 

23709 

25457 

25894 

20332 

1  6° 

22667 

23111 

23556 

24000 

24444 

2(822 

215G67 

27111 

380  XIX.— CUBIC  YARDS  PER  100  FEET.     SLOPES 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Ba-e 
28 

Base 
30 

Base 
32 

1 

52 

59 

67 

74 

81 

111 

119 

126 

2 

119 

133 

148 

163 

178 

237 

252 

267 

3 

200 

222 

244 

267 

289 

378 

400 

422 

4 

296 

326 

356 

385 

415 

533 

563 

£93 

5 

407 

444 

481 

519 

556 

704 

741 

778 

6 

533 

578 

622 

667 

711 

889 

933 

978 

7 

674 

726 

778 

830 

881 

1089 

1141 

1193 

8 

830 

889 

948 

1007 

1067 

1304 

1363 

1422 

9 

1000 

1067 

1133 

1200 

1267 

1533 

1600 

1667 

10 

1185 

1259 

1333 

1407 

1481 

1778 

1852 

1926 

11 

1385 

1467 

1548 

1630 

1711 

2037 

2119 

2200 

12 

1600 

1689 

1778 

1867 

1956 

2311 

2400 

2489 

13 

1830 

1926 

2022 

2119 

2215 

2600 

2696 

2793 

14 

2074 

2178 

2281 

2385 

2489 

2904 

3007 

3111 

15 

2333 

2444 

2556 

2667 

2778 

3222 

3:333 

3444 

16 

2607 

2726 

2844 

2963 

3081 

3556 

3674 

3793 

17 

2896 

3022 

3148 

3274 

3400 

3904 

4030 

4156 

18 

3200 

8333 

3467 

3600 

3733 

4267 

4400 

4533 

19 

3519 

3659 

3800 

3941 

4081 

4644 

4785 

4926 

20 

3852 

4000 

4148 

4296 

4444 

5037 

5185 

5333 

21 

4200 

4356 

4511 

4667 

4822 

5444 

5600 

5756 

22 

4563 

4730 

4889 

5052 

5215 

5867 

6030 

6193 

23 

4941 

5111 

5281 

5452 

5622 

6304 

6474 

6644 

24 

5333 

5511 

5689 

5867 

6044 

6756 

6933 

7111 

25 

5741 

5928 

6111 

6296 

6481 

7222 

7407 

7593 

26 

6163 

6356 

6548 

6741 

6933 

7704 

7896 

8089 

27 

6600 

6800 

7000 

7200 

7400 

8200 

8400 

8600 

28 

7052 

7259 

7467 

7674 

7881 

8711 

8919 

9126 

29 

7519 

7733 

7948 

8163 

8378 

9237 

9452 

9667 

30 

8000 

8222 

8444 

8667 

8889 

9778 

10000 

10222 

31 

8496 

872G 

8956 

9185 

9415 

10333 

10563 

10793 

32 

9007 

9244 

9481 

9719 

9956 

10904 

11141 

11378 

33 

9533 

9778 

10022 

10267 

10511 

11489 

11733 

11978 

34 

10074 

10326 

10578 

10830 

11081 

12089 

12341 

12593 

35 

10630 

10889 

11148 

11407 

11667 

12704 

12963 

13222 

36 

11200 

11467 

11733 

12000 

12267 

13.333 

13600 

13867 

37 

11785 

12059 

12333 

12607 

12881 

13978 

14252 

14526 

38 

12385 

12667 

12948 

13230 

13511 

14637 

14919 

15200 

39 

13000 

13289 

13578 

13867 

14156 

15311 

15600 

15889 

40 

13630 

13926 

14222 

14519 

14815 

16000 

16296 

16593 

41 

14274 

14578 

14881 

15185 

15489 

16704 

17007 

17311 

42 

14033 

15244 

15556 

15867 

16178 

17422 

17733 

18044 

43 

15607 

15926 

16224 

16563 

16881 

18156 

18474 

18793 

44 

16296 

16622 

16948 

17274 

17600 

18904 

19230 

19556 

45 

17000 

17333 

1766T 

1HOOO 

18333 

19667 

20000 

20333 

46 

17719 

18059 

18400 

18741 

19081 

20444 

20785 

21126 

47 

18452 

18800 

19148 

19496 

19844 

21237 

21585 

21933 

48 

19200 

19556 

19911 

20267 

20622 

22044 

22400 

22756 

49 

19963 

20326 

20689 

21052 

21415 

22867 

23230 

'23593 

50 

20741 

20711 

21481 

21852 

22222 

23704 

24074 

24444 

51 

21.  '33 

21911 

22289 

22667 

23044 

24556 

24933 

25311 

*2 

22341 

22726 

23111 

23496 

23881 

25422 

25807 

26193 

53 

23103 

23556 

23948 

24341 

24733 

26304 

26696 

27089 

54 

24000 

«  1400 

24800 

252(10 

2r)600 

27200 

27600 

28000 

55 

24852 

25259 

25667 

26074 

264S1 

28111 

28519 

28926 

56 

25719 

26133 

26548 

26963 

27378 

29037 

29452 

29867 

57 

25600 

27022 

27444 

27867 

28289 

20978 

30400 

30822. 

58 

27496 

27!>26 

28356 

28785 

29215 

30933 

31363 

31793 

59 

28407 

28844 

29281 

29719 

30156 

31904 

32341 

32778 

60 

29333 

2977'S 

30222 

30667 

31111 

32889 

33333 

33778 

XIX.— CUBIC  YARDS  PER  100  FEET.     SLOPES  3  :  1.  381 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

20 

28 

30 

32 

1 

56 

63 

70 

78 

85 

115 

122 

130 

2 

U33 

148 

163 

178 

193 

252 

267 

281 

3 

233 

256 

278 

300 

322 

411 

433 

456 

4 

356 

385 

415 

444 

474 

693 

622 

652 

5 

500 

537 

574 

611 

<U8 

796 

833 

870 

6 

667 

711 

756 

800 

844 

1022 

1067 

1111 

7 

856 

907 

959 

1011 

1063 

1270 

1322 

1374 

8 

1067 

1126 

1185 

1244 

1304 

1541 

1600 

1659 

9 

1300 

1367 

14:» 

1500 

1567 

1833 

1900 

1967 

10 

1556 

1630 

1704 

1778 

1852 

2148 

2222 

2296 

11 

1833 

1915 

1996 

2078 

2159 

2485 

2567 

2648 

12 

2133 

2222 

2311 

2400 

2489 

2844 

2933 

3022 

13 

2456 

2552 

2648 

2744 

2841 

3226 

3322 

3419 

14 

2800 

2904 

3007 

3111 

3215 

3630 

3733 

3837 

15 

3167 

3278 

3389 

3500 

3611 

4056 

4167 

4278 

16 

3556 

3674 

3793 

3911 

4030 

4504 

4622 

4741 

17 

3967 

4093 

4219 

4344 

4470 

4974 

5100 

5226 

18 

4400 

4533 

4667 

4800 

4933 

5467 

5600 

5733 

19 

4856 

4996 

5137 

5278 

5419 

5981 

6122 

6263 

20 

5333 

5481 

5030 

5778 

5926 

6519 

6667 

6815 

21 

5833 

5989 

6144 

6300 

6456 

7078 

7233 

7389 

22 

6356 

6519 

6681 

6844 

7007 

7659 

7822 

7985 

23 

6900 

7070 

7241 

7411 

7581 

8263 

8433 

8504 

24 

7467 

7644 

7822 

8000 

8178 

8889 

9067 

9144 

25 

8056 

8241 

8426 

8611 

8796 

9537 

9722 

9807 

26 

8667 

8859 

9052 

9244 

9437 

10207 

10400 

10593 

27 

9300 

9500 

9700 

9900 

10100 

10900 

11100 

11300 

28 

9956 

10163 

10370 

10578 

10785 

11615 

11822 

12030 

29 

10633 

10848 

11063 

11278 

11493 

12352 

12567 

12781 

30 

11333 

11556 

11778 

12000 

12222 

13111 

13333 

13556 

31 

12056 

12285 

12515 

12744 

12974 

13893 

14122 

14352 

32 

12800 

13037 

13274 

13511 

13748 

14696 

14933 

15170 

33 

13567 

13811 

14056 

14300 

14514 

15522 

15767 

16011 

34 

14356 

14607 

14859 

15111 

15363 

16370 

16622 

16874 

35 

15167 

15426 

15685 

15944 

16204 

17241 

17500 

17759 

36 

16000 

16267 

16533 

16800 

17067 

18138 

18400 

18667 

37 

16856 

17130 

17404 

17678 

17952 

19048 

19322 

19596 

38 

17733 

18015 

18296 

18578 

18859 

19985 

20267 

20548 

39 

18633 

18922 

19211 

19500 

19789 

20944 

21233 

21522 

40 

19556 

19852 

20148 

20444 

20741 

21926 

22222 

22516 

41 

20500 

20804 

21107 

21411 

21715 

22930 

23233 

23537 

42 

21467 

21778 

22089 

22400 

22711 

23956 

24267 

24578 

43 

22456 

22774 

23093 

23411 

23730 

25004 

25322 

25641 

44 

23467 

23793 

24119 

24444 

24770 

26074 

26400 

26726 

45 

24500 

24833 

25167 

25500 

25833 

27167 

27500 

27833 

46 

25556 

25896 

26237 

26578 

26919 

28281 

28622 

28963 

47 

26633 

26981 

27330 

27678 

28026 

29419 

29767 

30115 

48 

27733 

28089 

28444 

28800 

29156 

30578 

30933 

31289 

49 

28856 

29219 

29581 

29944 

30307 

31759 

32122 

32485 

50 

30000 

30370 

30741 

31111 

31481 

32963 

33333 

33704 

51 

31167 

31544 

31922 

32300 

32678 

34189 

34567 

34944 

52 

32356 

32741 

33126 

33511 

33896 

85487 

35822 

36207 

53 

33567 

33959 

34352 

34744 

35137 

36707 

37100 

37493 

54 

34800 

35200 

35600 

36000 

36400 

38000 

38400 

38800 

55 

36056 

36463 

36870 

37278 

37685 

39315 

39722 

40130 

56 

37333 

37748 

38163 

38578 

38993 

40652 

41067 

41481 

57 

38633 

39056 

39478 

39900 

40322 

42011 

42433 

42856 

58 

39956 

40385 

40815 

41244 

41674 

43393 

43822 

44252 

59 

41300 

41737 

42174 

42611 

43048 

44796 

45233 

45670 

60 

42667 

43111 

43556 

44000 

44444 

46222 

46667 

47111 

382  TABLE  XX.-— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 
Sq. 
Ft. 

Cubic 
Yards. 

Area. 

& 

Cubic 
Yards. 

Area  . 

I?: 

Cubic 
Yards. 

1 

3.7 

51 

188.9 

101 

374.1 

151 

559.3 

201 

744.4 

2 

7.4 

52 

192.6 

102 

377.8 

152 

563.0 

202 

748.2 

3 

11.1 

53 

196.3 

103 

381.5 

153 

566.7 

203 

751.9 

4 

14.8 

54 

200.0 

104 

385.2 

1     154 

570.4 

204 

755.6 

5 

18.5 

55 

203.7 

105 

388.9 

155 

574.1 

205 

759.3 

6 

22.2 

56 

207.4 

106 

392.6 

156 

577.8 

206 

763.0 

7 

25.9 

57 

211.1 

107 

390.3 

157 

581.5 

207 

766.7 

8 

29.6 

58 

214.8 

108 

4000 

158 

585.2 

208 

770.4 

9 

33.3 

59 

218.5 

109 

403.7 

159 

588.9 

209 

774.1 

10 

37.0 

60 

222.2 

110 

407.4 

160 

592.6 

210 

777.8 

11 

40.7 

61 

225.9 

111 

411.1 

161 

596.3 

211 

781  5 

Iti 

44.4 

62 

229.6 

112 

414.8 

162 

600.0 

212 

785.2 

13 

48.1 

63 

233.3 

113 

418.5 

163 

603.7 

213 

788.9 

14 

51.9 

64 

237  0 

114 

422  2 

164 

607.4 

214 

792.6 

15 

55.6 

65 

240.7 

115 

425^9 

165 

611.1 

215 

796.3 

16 

59.3 

66 

244.4 

116 

429.6 

166 

614.8 

216 

800.0 

17 

63.0 

67 

248.2 

117 

433.3 

167 

618.5 

217 

803.7 

18 

66.7 

68 

251.9 

118 

437.0 

168 

622.2 

218 

807.4 

19 

70.4 

69 

255.6 

119 

440.7 

169 

625.9 

219 

811.1 

20 

74.1 

70 

259.3 

120 

444.4 

170 

629.6 

220 

814.8 

21 

77.8 

71 

263.0 

121 

448.2 

171 

633.3 

221 

818.5 

22 

81.5 

72 

266.7 

122 

451.9 

172 

637.0 

222 

822.2 

23 

85.2 

73 

270.4 

123 

455.6 

173 

640.7 

223 

825.9 

24 

88.9 

74 

274.1 

124 

459.3 

174 

644.4 

224 

829.6 

25 

92.6 

75 

277.8 

125 

463.0 

175 

648.2 

225 

833.3 

28 

96.3 

76 

281.5 

126 

466.7 

176 

651.9 

226 

837.0 

27 

100.0 

77 

285  2 

127 

470.4 

177 

655.6 

227 

840.7 

28 

103.7 

78 

288.9 

128 

474  1 

178 

659.3 

228 

844.4 

29 

107.4 

79 

292.6 

129 

477.8 

179 

663.0 

229 

848.2 

30 

111.1 

80 

296.3 

130 

481.5 

180 

666.7 

230 

851.9 

31 

114.8 

81 

300.0 

131 

485.2 

181 

670.4 

231 

855.6 

32 

118.5 

82 

303.7 

132 

488  9 

182 

674.1 

282 

859.3 

33 

122.2 

83 

307.4 

133 

492.6 

1H3 

677.8 

233 

863.0 

34 

125.9 

84 

311.1 

134 

496,8 

184 

681.5 

234 

866.7 

35 

129  6 

85 

314.8 

135 

500.0 

185 

685  2 

235 

870.4 

36 

133.3 

86 

318.5 

136 

503  7 

186 

688.9 

236 

874.1 

37 

137.0 

87 

322.2 

137 

507.4 

187 

692.6 

237 

877.8 

38 

140.7 

88 

325.9 

138 

511.1 

188 

696.3 

238 

881.5 

39 

144  4 

89 

329.6 

139 

514.8 

189 

700.0 

239 

885.2 

40 

148.2 

90 

333.3 

140 

518.5 

190 

703.7 

240 

888.9 

41 

151.9 

91 

387.0 

141 

522.2 

191 

707.4 

241 

892.6 

42 

155.6 

92 

340.7 

142 

525.9 

192 

711.1 

242 

896.3 

43 

159.3 

93 

344.4 

143 

529.6 

193 

714.8 

243 

900.0 

44 

163.0 

94 

348.2 

[     144 

533  3 

194 

718.5 

244 

903.7 

45 

166.7 

95 

351.9 

145 

537.0 

195 

722.2 

245 

907.4 

46 

170.4 

96 

355.6 

146 

540.7 

196 

725.9 

246 

911.1 

47 

174.1 

97 

359.3 

147 

544.4 

197 

729.6 

247 

914.8 

48 

177.8 

G8 

363.0 

148 

548.2 

198 

733.3 

248 

918.5 

49 

181.5 

99 

366.7 

149 

551.9 

199 

737.0 

249 

922  2 

50 

185.2 

100 

370.4 

150 

555.6 

200 

740.7 

250 

925!  9 

TABLE  XX.— CUBIC  YARDS  IN  100  FEET  LENGTH.  383 


Area 

ft 

Cubic 
Yards. 

Area 

9k 

Cubic 
Yards. 

Area 

% 

Cubic 
Yards. 

Area 

9k 

Cubic 
Yards. 

Area. 
Sq. 
Ft. 

Cubic 
Yards. 

251 

929.6 

301 

1114.8 

351 

1300.0 

401 

1485.2 

451 

1670.4 

252 

933.3 

302 

1118.5 

352 

1303.7 

402 

1488.9 

152 

1674.1 

253 

937.0 

303 

1122.2 

353 

1307.4 

403 

1492.6 

453 

1  677  .  8 

254 

940.7 

304 

1125.9 

354 

1311.1 

404 

1496.3 

454 

1681.5 

255 

944.4 

305 

1129.6 

355 

1314.8 

405 

1500.0 

455 

1685.2 

266 

948.2 

306 

1133.3 

356 

1318.5 

406 

1503.7 

456 

1688.9 

257 

951.9 

307 

1137.0 

357 

1322.2 

407 

1507.4 

»57 

1692.6 

258 

955.6 

308 

1140.7 

358 

1325.9 

408 

1511.1 

458 

1696.3 

25!) 

959.3 

309 

1144.4 

359 

1329.6 

409 

1514.8 

459 

1700.0 

2(50 

963.0 

310 

1148.2 

360 

1333.3 

410 

1518.5 

460 

1703.7 

261 

966.7 

311 

1151.9 

361 

1337.0 

411 

1522  2 

461 

1707.4 

262 

970.4 

312 

1155.6 

362 

1340.7 

412 

1525.9 

41)2 

1711.1 

203 

974.1 

313 

1159.3 

363 

1344.4 

413 

1529.6 

401 

1714.8 

26  1 

977.8 

314 

1163.0 

364 

1348.2 

414 

1533.3 

464 

1718.5 

265 

-981.5 

315 

1166.7 

365 

1351.9 

415 

1537.0 

465 

1722.2 

266 

985.2 

316 

1170.4 

366 

1355.6 

416 

1540.7 

466 

1725.9 

267 

988.9 

317 

1174.1 

367 

1359.3 

417 

1544.4 

467 

1729.6 

268 

992.6 

318 

1177.8 

368 

1363.0 

418 

1548.2 

468 

1733.3 

26!) 

996.3 

319 

1181.5 

369 

1366.7 

419 

1551.9 

469 

1737.0 

270 

1000.0 

320 

1185.2 

370 

1370.4 

420 

1555.6 

470 

1740.7 

271 

1003.7 

321 

1188.9 

371 

1374.1 

421 

1559.3 

471 

1744.4 

272 

1007.4 

322 

1192.6 

372 

1377.8 

422 

1563.0 

472 

1748.2 

273 

1011.1 

323 

1196.3 

373 

1381.5 

423 

1566.7 

473 

1751.9 

274 

1014.8 

324 

1200.0 

374 

1385.2 

424 

1570.4 

474 

1755.6 

275 

1018.5 

325 

1203.7 

375 

1388.9 

425 

1574.1 

475 

1759.3 

276 

1022.2 

326 

1207.4 

376 

1392.6 

426 

1577.8 

476 

1763.0 

277 

1025.9 

327 

1211.1 

377 

1396.3 

427 

1581.5 

477 

1766.7 

rt-'Q 

1029.6 

328 

1214.8 

378 

1400.0 

428 

1585.2 

478 

1770.4 

279 

1033  3 

329 

1218.5 

379 

M03.7 

429 

1588.9 

479 

1774.1 

280 

1037.0 

330 

1222  2 

380 

1407.4 

430 

1592.6 

480 

1777.8 

281 

1040.7 

331 

1225.9 

381 

1411.1 

431 

1596.3 

481 

1781.5 

:?8-2 

1044.4 

332 

1229  .6 

382 

1414.8 

432 

1600.0 

482 

1785.2 

283 

1048.2 

333 

1233.3 

383 

1418.5 

433 

1603.7 

483 

1788  9 

284 

1051.9 

334 

1237.0 

384 

1422.2 

434 

1607.4 

484 

1792.6 

285 

1055.6 

335 

1240.7 

385 

1425.9 

4% 

1611.1 

485 

1796.3 

286 

1059.3 

336 

1244.4 

386 

1429  6 

436 

1614.8 

486 

1800.0 

287 

1003.0 

337 

1248.2 

387 

1433.3 

437 

1618.5 

487 

1803.7 

288 

1066.7 

338 

1251.9 

388 

1437.0 

438 

1622.2 

488 

1807.4 

289 

?070.4 

339 

1255.6 

389 

1440.7 

439 

1625.9 

189 

1811.1 

290 

1074.1 

340 

1259.3 

390 

1444.4 

440 

1629.6 

490 

1814.8 

291 

1077.8 

311 

1263.0 

391 

1448.2 

441 

1633.3 

491 

1818.5 

292 

1081.5 

3*2 

1266.7 

392 

1451.9 

442 

1687.0 

492 

1822.2 

293 

1085.2  : 

343      1270.4 

393 

1455.6 

443 

1640.7 

493 

1825.9 

294 

1088.9  I 

344      1274.1 

394 

1459.3 

444  i   1644.4 

494 

1829.6 

295 

1092.6 

345      1277.8 

395 

1463.0 

445 

1648  2 

495 

1833.3 

296 

1096.3 

346      1-281.5 

396 

1466.7 

4l(i 

1651.9 

496 

1837.0 

297      1100.0 

347      1285.2 

397 

1470.4 

447      1655.6 

497 

1840.7 

298      1103.7 

348 

1288.9 

398 

1474.1 

448 

1659.3 

498 

1844.4 

299 

1107.4 

349 

1292.6 

399 

1477.8 

449      1663.0 

499 

1848.2 

300 

1111.1 

350 

1296.3 

400      1481.5 

450 

1066.7  ! 

500 

1851.9 

384  TABLE  XX.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 
Sq. 
Ft. 

Cubic 
Yards. 

Area. 

£ 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

§* 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards-. 

501 

1855.6 

551 

2040.7 

601 

2225.9 

651 

2411.1 

701 

2596.3 

502 

1859.3 

552 

2044.4 

602 

2229.6 

652 

2414.8 

702 

2600.0 

503 

1863.0 

553 

2048.2 

603 

2233.3 

653 

2418.5 

703 

2603.7 

504 

1866.7 

554 

2051.9 

604 

2237.0 

654 

2422.2 

704 

2607.4 

505 

1870.4 

555 

2055.6 

605 

2240.7 

655 

2425.9 

705 

2611.1 

506 

1874.1 

556 

2059.3 

606 

2244  .4 

656 

2429.6 

706 

2614.8 

507 

1877.8 

557 

2063.0 

607 

2248.2 

657 

2433.3 

707 

2618.5 

508 

1881.5 

558 

2066.7 

608 

2251.9 

658 

2437.0 

708 

2622.2 

509 

1885.2 

559 

2070.4 

609 

2255.6 

659 

2440.7 

709 

2625.9 

510 

1888.9 

560 

2074.1 

610 

2259.3 

660 

2444.4 

710 

2629  6 

511 

1892.6 

561 

2077.8 

611 

2263.0 

661 

2448.2   ! 

711 

2633.3 

512 

1896.3 

562 

2081.5 

612 

2266.7 

662 

2451.9   i 

712 

2637.0 

513 

1900.0 

563 

2085.2 

613 

2270.4 

663 

2455.6 

713 

2640.7 

514 

1903.7 

564 

2088.9 

614 

2274.1 

664 

2459.3   | 

714 

2644.4 

515 

1907.4 

565 

2092.6 

615 

2277.8 

665 

2463.0   i 

715 

2648.2 

5)6 

1911.1 

566 

2096.3 

616 

2281.5 

666 

2466.7 

716 

2651.9 

517 

1914.8 

567 

2100.0 

617 

2285.2 

667 

2470.4 

717 

2655.6 

518 

1918.5 

568 

2103.7 

618 

2288.9 

668 

2474.1 

718 

2659.3 

51  9 

1922.2 

569 

2107.4 

619 

2292.6 

669 

2477.8 

719 

2663.0 

520 

1925.9 

570 

2111.1 

620 

2296.3 

670 

2481.5 

720 

2666.7 

521 

1929  6 

571 

2114.8 

621 

2300.0 

671 

2485.2 

721 

2670.4 

522 

1933.3 

572 

2118.5 

622 

2303.7 

672 

2488.9 

722 

2674.1 

523 

1937.0 

573 

2122.2 

623 

2307.4 

673 

2492  6 

723 

2677.8 

524 

1940.7 

574 

2125.9 

624 

2311.1 

674 

2496.3 

724 

2681.5 

525 

1944.4 

575 

2129.6 

625 

2314.8 

675 

2500.0 

725 

2685.2 

526 

1948.2 

576 

2133.3 

626 

2318.5 

676 

2503.7 

726 

2688.9 

52? 

1951.9 

577 

2137.0 

627 

2322.2 

677 

2507.4 

727 

2692.6 

528 

1955.6 

578 

2140.7 

628 

2325.9 

678 

2511.1 

728 

2696.3 

529 

1959.3 

579 

2144.4 

629 

2329.6 

679 

2514.8 

729 

2700.0 

530 

1963.0 

580 

2148  2 

630 

2333.3 

680 

2518.5 

730 

2703.7 

531 

1966.7 

581 

2151.9 

631 

2337.0 

681 

2522.2 

731 

2707.4 

532 

1970.4 

582 

2155.6 

632 

2340.7 

682 

2525.9 

732 

2711.1 

533 

1974.1 

583 

2159.3 

633 

2344.4 

683 

2529.6 

733 

2714.8 

534 

1977.8 

584 

2163.0 

634 

2348.2 

684 

2533.3 

7J34 

2718.5 

535 

1981.5 

585 

2166.7 

635 

2351.9 

685 

2537.0 

735 

2722.2 

536 

1985.2 

586 

2170.4 

636 

2355.6 

686 

2540  7 

736 

2725.9 

537 

1988.9 

587 

2174.1 

637 

2359.3 

687 

2544.4 

737 

2729.6 

538 

1992.6 

588 

2177.8 

638 

2363  0 

688 

2548.2 

738 

2733.3 

539 

1996.3 

589 

2181.5 

639 

2366.7 

689 

2551.9 

739 

2737.0 

540 

2000.0 

590 

2185.2 

640 

2370.4 

690 

2555.6 

740 

2740.7 

541 

2003.7 

591 

2188.9 

641 

2374.1 

691' 

2559.3 

741 

2744.4 

542 

2007.4 

592 

2192.6 

642 

2377.8 

692 

2563.0 

742 

2748.2 

543 

2011.1 

593 

2196.3 

643 

2381.5 

693 

2566.7 

743 

2751.9 

544 

2014.8 

594 

2200.0 

644 

2385.2 

694 

2570.4 

744 

2755.6 

545 

2018.5 

595 

2203.7 

645 

2388.9 

695 

2574.1 

745 

2759.3 

546 

2022.2 

596 

2207.4 

646 

2392.6 

696 

2577.8 

746 

2763.0 

547 

2025.9 

597 

2211.1 

647 

2396.3 

697 

2581.5 

747 

2766.7 

548 

2029.6 

598 

2214.8 

648 

2400.0 

698 

2585.2 

748 

2770.4 

549 

20:33.3 

599 

2218.5 

649 

2403.7 

699 

2588.9 

749 

2774.1 

550 

2037.0 

600 

2222.2 

650 

2407.4 

700 

2592.6 

750 

2777.8 

XX.— CUBIC  YARDS  IN  100  FEET  LENGTH.  385 


Area. 

"ft 

Cubic 
Yards. 

A  rea  . 
Sq. 
Ft. 

Cubic 
Yards. 

Area. 

!?: 

Cubic 
Yards. 

Area. 

1?: 

Cubic 

Yards. 

Area. 

ft 

Cubic 
Yards. 

751 

2781.5 

801 

2966.7 

851 

3151.9 

901 

3337.0 

951 

3522.2 

752 

2785.2 

802 

2970.4 

852 

3155.6 

902 

3340.7 

952 

3525.9 

753 

2788.9 

803      2-J74.1 

853 

3159.3   ! 

903 

3344.4 

953 

3529.6 

754 

2792.6 

804      2977.8 

854 

3163.0 

904 

3348.2 

954 

3533.3 

755 

2796.3 

805      2981.5 

855 

3166.7 

905 

3351.9 

955 

3537.0 

756 

2800.0 

806      2985.2 

856 

3170.4 

906 

3355.6 

956 

3540.7 

757 

2803.7 

807     2988.9 

857  i  3174.1 

907 

3359.3 

957 

3544.4 

758 

2807.4 

808 

2992.6 

858  \  3177.8 

908 

3363.0 

958 

3548.2 

759 

2811.1 

809 

2996.3 

859 

3181.5 

909 

3366.7 

959 

3551.9 

760 

2814.8 

810 

31)00.  0 

860 

3185.2 

910 

3370.4 

960 

3555.6 

761 

2818.5 

811 

3003  7 

861 

3188.9 

911 

3374.1 

961 

3559.3 

762 

2822.2 

812 

3007.4 

862 

3192.6 

912 

3377.8 

962 

3563.0 

763 

2825.9 

813 

3011.1 

863 

3196.3   : 

913 

3381.5 

963 

3566.7 

764 

2829/6 

814 

3014.8 

864 

3200.0 

914 

3385.2 

964 

3570.4 

765 

2833  3 

815 

3018.5 

865 

3203.7 

915 

3388.9 

965 

3574.1 

766 

2837.0 

816 

3022.2 

866 

3207.4 

916 

3392.6 

966 

3577.8 

767 

2840.7 

817 

3025.9 

867 

3211.1 

917 

3396.3 

967 

3581.5 

768 

2844.4 

818 

3029.6 

868 

3214.8 

918 

3400.0 

968 

3585.2 

769 

2848.2 

819 

3033.3 

869 

3218.5 

919 

3403.7 

969 

3588.9 

770 

2851.9 

820 

3037.0 

870 

3222.2 

920 

3407.4 

970 

3592.6 

771 

2855.6 

821 

3040.7 

871 

3225.9 

921 

3411.1 

971 

3596.3 

772 

2859.3 

822 

3044.4 

872 

3229.6 

922 

3414.8 

972 

3600.0 

773 

2863.0 

823 

3048.2 

873 

3233.3 

923 

3418.5 

973 

3603.7 

774 

2866.7 

824 

3051.9 

874 

3237.0 

924 

3422.2 

974 

3607.4 

775 

2870.4 

825 

3055.6 

875 

3240.7 

925 

3425.9 

975 

3611.1 

776 

2874.1 

826 

3059.3 

876 

3244.4 

926 

3429.6 

976 

3614.8 

777 

2877.8 

827 

3063.0 

877 

3248.2 

927 

3433.3 

977 

3618.5 

778 

2881.5 

828 

3066.7 

878 

3251.9   i 

928 

3437.0 

978 

3622.2 

779 

2885.2 

829 

3070.4 

879 

3255.6 

929 

3440.7 

979 

3625.9 

780 

2888.9 

830 

3074.1 

880 

3259.3 

930 

3444.4 

980 

3629.6 

781 

2892.6 

831 

3077.8 

881 

3263.0 

931 

3448.2 

981 

3633.3 

782 

2896.3 

832 

3081.5 

882 

3266.7 

932 

3451.9 

982 

3637.0 

783 

2900.0 

833 

3085.2 

883 

3270.4 

933 

3455.6 

983 

3640.7 

784 

2903.7 

834 

3088.9 

884 

3274.1 

934 

3459.3 

984 

3644.4 

785 

2907.4 

835 

3092.6 

885 

3277.8 

935 

3463.0 

985 

3648.2 

786 

3911.1 

836 

3096.3 

886 

3281.5 

936 

3466.7 

986 

3651.9 

787 

2914.8 

837 

3100.0 

887 

3285.2 

937 

3470.4 

987 

3655.6 

788 

2918.5 

838 

3103.7 

888 

3288.9 

938 

3474.1 

988 

3659.3 

789 

2922.2 

839 

3107.4 

889 

3292.6 

939 

3477.8 

Q89 

3663.0 

790 

2925.9 

840 

3111.1 

890 

3296.3 

940 

3481.5 

990 

3666.7 

791 

2929.6 

841 

3114.8 

891 

3300.0 

941 

3485.2 

991 

3670.4 

792 

2933.3 

842 

3118.5 

892 

3303.7 

942 

3488.9 

992 

3674.1 

793 

2937.0 

843 

3122.2 

893 

3307.4 

943 

3492.6 

993 

3677.8 

794 

2940.7 

844 

3125.9 

894 

3311.1 

944 

3496.3 

994 

3681.5 

795 

2944.4 

845 

3129.6 

895 

3314.8 

945 

3500.0 

995 

3685.2 

796 

2948.2 

846 

3133.3 

896 

3318.5 

946 

3503.7 

996 

3688.9 

797 

2951.9 

847 

3137.0 

897 

3322.2 

947 

3507.4 

997 

3692.6 

798       2955.6 

848 

3140.7 

898 

3325.9 

948 

3511.1 

998 

3696.3 

799 

2959.3 

849 

3144.4 

899 

3329.6 

949 

3514.8 

999 

3700.0 

800 

2963.0 

850 

3148.2 

900 

3333.3   I 

950 

3518.5 

1000 

3703.7 

386       XXI.— RISE  PER  MILE  OF  VARIOUS  GRADES. 


Rise 
Cent. 

Feet  pei- 
Mile. 

Rise 
per 

Cent. 

Feet  pei- 
Mile. 

Rise 
pei- 
Cent. 

Feet  pei- 
Mile. 

Rise 
per 
Cent. 

'    ~j 
Feet  per 
Mile. 

.01 

.528 

.61 

32.208 

1.21 

63.888 

1.81 

95.568 

.02 

1.056 

.62 

32.736 

1.22 

64.416 

1.82 

96.096 

.03 

1.584 

.63 

33.264 

1.28 

64.944 

1.83 

96.6:24 

.04 

2.112 

.64 

33.792 

1.24 

65.472 

1.84 

97.152 

.05 

2.640 

.65 

34.320 

1.25 

66.000 

1.85 

97.680 

.06 

3.168 

.66 

34.848 

1.26 

66.528 

1.86 

98.208 

.07 

3.696 

.67 

35.376 

1.27 

67.056 

1.87 

98.736     , 

.08 

4.224 

.68 

35.904 

1.28 

67.584 

1.88 

99.264 

.09 

4.752 

.69 

36.43'2 

1.29 

08.112 

1.89 

99.792 

.10 

5.280 

.70 

36.9GO 

1.30 

68.640 

1.90 

100.320 

.11 

5.808 

.71 

37.488 

1.31 

69.168 

1.91 

100.848 

.12 

6.336 

.72 

38.016 

1.32 

69.696 

1.92 

101.376 

.13 

6.864 

.73 

38.544 

1.33 

70.224 

1.93 

101.904 

.14 

7.392 

.74 

39.072 

1734 

70.752 

1.94 

102.432 

.15 

7.920 

.75 

SO.  600 

1.35 

71.280 

1.95 

102.960 

.16 

8.448 

.76 

40.128 

1.36 

71.808 

1.96 

103.488 

.17 

8.976 

.77 

40.656 

1.37 

72.336 

1.97 

104.016 

.18 

9.504     ; 

.78 

41.184 

1.38 

72.864 

1.98 

104.544 

.19 

10.032 

.79 

41.712 

1.39 

73.392 

1.99 

105.072 

.20 

10.500 

.80 

42.240 

1.40 

73.920 

2.00 

105.600 

.21 

11.088 

.81 

42.768 

1.41 

74.448 

2.10 

110.880 

.22 

11.616 

.82 

43.296 

1.42 

74.976 

2.20 

116.160 

.23 

12.144 

.83 

43.824 

1.43 

75.504 

2.30 

121.440 

.24 

12.672 

.84 

44.352 

1.44 

76.032 

2.40 

120.720 

.25 

13.200 

.85 

44.880 

1.45 

76.560 

2.50 

132.000 

.26 

13.728 

.86 

45.408 

1.46 

77.088 

2.60 

137.280 

.27 

14.256 

.87 

45.936 

1.47 

77.616 

2.70 

142.560 

.28 

14.784 

.88 

46.464 

1.48 

78.144 

2.80 

147.840 

.29 

15.312 

.89 

46.992 

1.49 

78.672 

2.90 

153.120 

.30 

15.840 

.£0 

47.520 

1.50 

79.200 

3.00 

158.400 

.31 

16.368 

.91 

48.048 

1.51 

79.728 

3.10 

163.680 

.32 

16.896 

.92 

48.576 

1.52 

80.256 

3.20 

168.960 

.33 

17.424 

.93 

49.104 

1.68 

80.784 

3.30 

174.240 

.34 

17.952 

.94 

49.632 

1.54 

81.312 

3.40 

179.520 

.35 

18.480 

.95 

.00.160 

1.55 

81.840 

3.50 

184.800 

.36 

1.9.008 

.96 

50.688 

1.56 

82.368 

3.60 

190.080 

.37 

19.536 

.97 

51.  *6 

1.57 

82.896 

3.70 

195.360 

.38 

20.064 

.98 

51  744 

1.58 

83.424 

3.80 

200.640 

.39 

20.592 

.99 

52.272 

1.59 

83.952 

3.90 

205.920 

.40 

21.120 

1.00 

52.800 

1.60 

84.480 

4.00 

211.200 

.41 

21.648 

1.01 

53.328 

1.61 

85.008 

4.10 

216.480 

.42 

22.176 

1.02 

53.856 

1.62 

85.536 

4.20 

221.760 

.43 

22.704 

1.03 

54.384 

1.63 

86.064 

4.30 

227.040 

.44 

23.232 

.04 

54.912 

1.64 

86.592 

4.40 

232.320 

.45 

23.760 

.05 

55.440 

1.65 

87.120 

4.50 

237.600 

.46 

24.288 

.06 

55.968 

1.66 

87.648 

4.60 

242.880 

.47 

24.816 

.07 

56.496 

1.67 

88.176 

4.70 

248.160 

.48 

25.344 

.08 

57.024 

1.68 

88.704 

4.80 

253.440 

.49 

25.872 

.09 

57.552 

1.69 

89.232 

4.90 

258.720 

.50 

26.400 

.10 

58.080 

1.70 

89.760 

5.00 

264.000 

51 

26.928 

.11 

58.608 

1.71 

90.288 

5.10 

269.280 

.52 

27.456 

.12 

59.136 

1.72 

90.816 

5.20 

274.560 

53 

27.984 

.13 

59.664 

1.73 

91.344 

5.30 

279.840 

.54 

28.512 

.14 

60.192 

1.74 

91.872 

5.40 

285.120 

.55 

29.040 

.15 

60.720 

1.75 

92.400 

5.50 

290.400 

.56 

29.568 

.16 

61.248 

1.76 

92.928 

5.60 

295.680 

.57 

30.096 

.17 

61.776 

1.77 

93.456 

5.70 

300.960 

.58 

30.624 

.18 

62.304 

1.78 

93.984 

5  80 

306.240 

.59 

31.152 

.19 

62.832 

1.79 

94.512 

5.90 

311.520 

.60 

31.680 

.20 

63.360 

1.80 

95.040 

6.00 

316.800 

TABLE   XXII.— SLOPES   FOR   TOPOGRAPHY. 


38r 


fl- 

«    _: 

Oo 

1        c 

a>     ~ 

00 

n 

ai    -; 

oo" 

1            Q 

v:       c3 

0 

.22     5 

-*->  *-< 

o 

•ji     - 

•*—  •—  » 

'£ 

S    § 

SI'S 

s 

P£      ~* 

tf     a 

«So 

^,s 

^Tj 

"So   N 

.2S'g 

Hi 

u_  cG 

O  S 

2? 

•SgJ 
£|i 

ljj 

"o  c 
£^ 

~     o 

cd  O.N 

5*  a 

*j~  J^s 

b£  = 

"S  •"  ^ 

'C  *^r 

C  >-5 

c^: 

c  •—  ' 

> 

K 

F 

ffi 

> 

K 

0°  20' 

.88 

1718.9 

7°  20' 

*2.87 

r.,-    ~ 

16° 

28.67 

34.9 

40 

1.16 

859.4 

40 

13^46 

74  .'3 

17 

30.57 

32.7 

1 

1.75 

572.9 

8 

14.05 

71.2 

18 

32.49 

30.8 

20 

2.83 

429.6 

20 

14.05 

68.3 

19 

34.43 

29.0 

40 

2.91 

343.7 

40 

35  24 

65.6 

20 

36.40 

27.5 

2 

3.4!) 

286.4 

9 

15.84 

63.1 

21 

38.39 

26.1 

20 

4.08 

245.4 

20 

16.44 

60.8 

22 

40.40 

24.8 

40 

4.00 

214.7 

40 

17.03 

58.7 

23 

42.45 

23.6 

3 

5.24 

190.8    ! 

10 

17.03 

56.7 

24 

44.52 

22.5 

20 

5.82 

171.7 

20 

18.23 

54.8 

25 

46.63 

21.4 

10 

6.41 

156.0 

40 

18.84 

53.1 

26 

48.77 

20.5 

4 

6.99 

143.0 

11 

19.44 

51.4 

27 

50.95 

19.6 

20 

7.58 

132.0 

30 

20.35 

49.2 

28 

53.17 

18.8 

40 

8.16 

122.5 

12 

21.26 

47.0 

29 

55.43 

18.0 

5 

8.75 

114.3 

30 

22.17 

45.1 

30 

57.74 

17.3 

20 

9.34 

107.1 

13 

23!  09 

43  3 

35 

70.02 

14.3 

40 

9.92 

100.8 

30 

24.01 

41.7 

40 

83.91 

11.9 

6 

10  51 

95.1 

14 

24.93 

40.1 

45 

100.00 

10.0 

20 

11.10 

90.1 

30 

25.86 

38.7 

50 

119.18 

8.4 

40 

11.69 

85.6 

15 

26.79 

37.3 

55 

142.81 

7.0 

1 

12.28 

81.4 

30 

27.73 

36.1 

60 

173.21 

5.8 

I 

TABLE  XXIII.— MATERIAL  REQUIRED  FOR  ONE   MILE  OF  TRACK. 


RAIL    WEIGHTS. 

RAILROAD   SPIKES. 

-O 

<" 

•  «4 

Required  for 

i 

03 
°   CO 

a  aj 

b 

SM 

Ties  2  ft.  Apart. 

O.O 

•c    - 

0)  0  to 

For  Rails 

|1 

So 

fl 

111 

|S 

o! 

Weighing 

PH 

s 

w 

CO 

<1 

i^  ••* 

^ 

12 

21.12 

18.857 

^XT9e 

360 

5870 

29.3 

45  to 

70  Ibs. 

16 

28.16 

25.14 

i 

5  XT9S 

400 

5280 

26.4 

40 

56 

20 

35.20 

31  .  42 

} 

5  X  i 

450 

4690 

23.5 

35 

40 

25 

44.00 

39.286 

530 

3980 

19.9 

28 

35 

30 

52.80 

47.14 

^ 

4  X  i 

600 

3520 

17.6 

24 

35 

35 

61.60 

55.00 

1 

680 

3110 

15.5 

20 

30          ; 

40 

70.40 

62.85 

4  X/B 

720 

2930 

14.7 

20 

30 

45 

79.20 

70.71 

1 

3i  y/  7 

900 

2350 

11.7 

16 

25 

56 

98.56 

88.00 

1 

4  X  f 

1000 

2110 

10.  fr 

16 

25 

60 

105.60 

94.28 

s 

34  X  f 

1190 

1770 

8.9 

16 

20 

70 

123.20 

110.000 

3  X  f 

1240          1700 

8.5 

16 

20 

80 

140.80 

125.714 

*»^      b 

1342 

1570 

7.9 

12 

16 

NUMBER   OF   SPLICE-JOINTS. 

NUMBER  OF  CROSS-TIES.              Two  Bars  with  Four  Bolts  and 

Nuts  to  Each  Joint. 

Distance  apart,  c.  to  c.,  in  Feet.                     Length  of  Rail  in  Feet. 

1.5          1.75 

2.0 

2.25       2.50          20 

24 

26 

28 

30 

3520        3017        2640 

2347       2112          528 

440 

406 

352 

388 


TABLE   XXIV. 


CONVERSION  OF  ENGLISH  INCHES  INTO  CENTIMETRES. 

Ins. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

0 

0.000 

2.540 

5.080 

7.620 

10.16 

12.70 

15.24 

17.78 

20.32 

22.86 

10 

25.40 

27.94 

30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

48.26 

20 

50.80 

53.34 

55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74     81.28 

83.82 

86.36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

104.14    106.68 

109.22 

111.76 

114.30 

116.84 

119.38 

121.92 

124.46 

50 

127.00 

129.54!  132.08 

134.62 

137.16 

139.70 

142.24 

144.78 

147.32 

149.86 

60 

152.40 

154.94    157.48 

160.02 

162.56 

165.10 

167.64 

17'0.18 

172.72 

175.26 

70 

177.80 

180.  34!  182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.96 

80 

203.20 

205.  74    208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.141  233.68 

236.22 

238.76 

241.30 

243.84 

246.38 

248.92 

251.46 

100 

254.00 

256.54    259.08 

261.62 

264.16 

266.70 

269.24271.78 

274.32 

276.8!) 

CONVERSION  OF  CENTIMETRES  INTO  ENGLISH  INCHES. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins.      Ins. 

Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787 

1.181 

1.575 

1.969    2.362    2.756 

3.150 

3.543 

10 

3.937 

4.331 

4.742 

5.118 

5.512 

5.906i  6.299    6.693 

7.087    7.480 

20 

7.874 

8.268 

8.662     9.055 

9.449 

9.843110.236  10.630 

11.  024|11.  418 

30 

11.811 

12.205 

12.599    12.992 

13.386 

13.780il4.1V314.567 

14.  961115.  355 

40 

15.748 

16.142 

16.536    16.929 

17.323 

17.717118.111  18.504 

18.89819.292 

50 

19.685 

20.079 

20.473    20.867 

21.260 

21.  654  '22.048  22.  441 

22.835  23.229 

60 

23.622 

24.016 

24.410    24.804 

25.197 

25.591  25.  985  26.  378 

26.77227.166 

70 

27.560 

27.953   28.347    28.741 

29.134 

29.52829.92230.316 

30.70931.103 

80 

31.497 

31.890    32.284    32.678 

33.071 

33.46533.85934.253 

34.64635.040 

90 

35.434 

35.827    36.221    36.615 

37.009 

37.402l37.79638.190 

38.58338.977 

100 

39.3701  39.764 

40.158    40.552 

40.945 

41.339J41.73342.126 

42.52042.914 

CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

0 

0.000 

0.3048 

0.6096 

0.9144 

1.2192 

1.52391.82872.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

3.6575 

3.9623 

4.2671 

4.57194.876715.1815 

5.4863 

5.7911 

20 

6.0359 

6.4006 

6.7055 

7.0102 

7.3150 

7.61987.92468.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.668 

10.97211.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.71614.020:14.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.76317.06817.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19.202 

19.507 

19.81120.11620.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.85923.16423.469 

23.774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.90726.21226.517 

26.822 

27.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.95529.26029.565 

29.870 

30.174 

100 

30.479 

30.784 

31.089 

31.394 

31.698 

32.  003132.  308  32.  613 

32.918|33.222 

CONVERSION  OF  METRES  INTO  ENGLISH   FEET. 

Met. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Feet. 

Feet. 

Feet.     Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5618    9.8427 

13.123 

16.404  19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.371:  42.651 

45.932 

49.213i52.494 

55.775 

59.056 

62.337 

20 

65.618 

68.899 

72.179    75.461 

78.741 

82.02285.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99    108.27 

111.55 

114.83118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80    141.08 

144.36 

147.  64  (150.92 

154.20 

157.48 

160.76 

50 

164.04 

167.33 

170.61    173.89 

177.17 

180.45^183.73 

1H7.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42    206.70 

209.98 

213.26216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22    239.51 

242.79 

246.07249.35 

252.63 

255.91 

259.19 

80 

262.47 

265.75 

269.03    272.31 

275.60 

278.88282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

391.84    305.12 

308.40 

311.69314.97 

318.25 

321.53 

324.81 

100 

328.09 

331.37 

334.65!  337.93 

341.21 

344.  49  1347.78 

351.06354.34 

357.62 

TABLE  xxV, 


CONVERSION  OF 

ENGLISH  STATUTE-MILES  INTO  KILOMETRES.     | 

Miles. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

0 

0.0000 

1.6093 

3.2186 

4.82796.4372    8.0465 

9.655811.2652 

12.8745 

14.4848 

10 

16.093 

17.70219.312 

20.921  22.530    24.139 

25.749   27.358 

28.967 

30.577 

20 

32.186 

33.795 

35.405 

37.014 

88.688 

40.232 

41.842 

43.451 

45.060 

46.670 

30       48.279 

49.888 

51.498 

53.107 

54.71€ 

56.325 

57.935 

59.544 

61.153 

62.763 

40      164.372  65.  981 

67.591 

69.200 

70.  80S 

72.418 

74.028 

75.637 

77.246 

78.856 

50      [80.465 

82.074 

83.684 

85.293 

86.90X 

88.511 

90.121 

91.730 

93.339 

94.949 

60       96.55898.16799.777 

101.39 

102.  9£ 

>    104.60 

106.21 

107.82    109.43 

111.04 

70 

112.65 

114.26 

115.87 

117.48 

119.  0£ 

$    120.69 

122.30 

123.91!  125.52 

127.13 

80 

128.74 

130.35 

131.96 

133.57 

135.  1' 

'    136.78 

138.39 

140.00    141.61 

143.22 

90 

144.85146.44148.05 

149.66 

i5i.se 

5    152.87 

154.48 

156.09    157.70 

159.31 

100 

160.93 

162.531164  14 

165  75 

167.  3c 

>    168.96 

170.57 

172.181  173.79 

175.40 

CONVERSION  OF 

KILOMETRES  INTO   ENGLISH   STATUTE-MILES. 

Kilom. 

0 

1 

2 

3 

4 

5 

6            7 

8 

9 

Miles. 

Miles. 

Miles. 

Miles. 

Miles 

.   Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

0 

0.0000 

0.6214 

1.2427 

1.8641 

2.485. 

>    3.1069 

3.7282    4.3497 

4.9711 

5.5924 

10 

6.2138 

6.8352 

7.4565 

8.0780 

8.699^ 

f    9.3208 

9.9421 

10.562 

11.185 

11.805 

20 

12.427 

13.049 

13.670 

14.292 

14.91, 

5    15.534 

16.156 

16.776 

17.399 

18.019 

30 

18.641 

19.263 

19.884 

20.506 

21.12' 

"    21.748 

23.370 

22.990 

23.613 

24.233 

40 

24.855 

25.477 

26.098 

26.720 

27.34 

27.962 

28.584 

29.204 

29.827 

30.447 

'     50 

31.069 

31.690 

32.311 

32.933 

33.55 

1    34.175 

34.797 

35.417 

36.040 

36  660 

60 

37.282 

37.904 

38.525 

39.147 

39.76* 

3    40.389 

41.011 

41.631 

42.254 

42.874 

70 

43.497 

44.118 

44.739 

45.361 

45.98 

3    46.603 

47.225 

47.845 

48.468 

49.088 

80 

49.711 

50.332 

50.953 

51.575 

52.19 

5    52.817 

53.439 

54.059 

54.682 

55.302 

90 

55.924 

56.54S 

57.166 

57.788 

58.40 

)    59.030 

59.652 

60.272 

60.895 

61.515 

100 

62.13862.759 

63.380 

64.002 

64.62 

i    65.244 

65.866 

66.486 

67.109 

67.729 

TABLE  XXVI. 

LENGTH  IN  FEET  OF  1'  ARCS  OF  LATITUDE  AND  LONGITUDE. 

Lat. 

1'  Lat. 

1'  Long. 

Lat. 

1'  Lat. 

T  Long. 

1° 

6045 

6085 

31° 

6061 

5222 

2° 

6045 

6083 

32° 

6062 

5166 

3° 

6045 

6078 

33° 

6063 

5109 

4° 

6045 

6071 

34° 

6064 

5051 

5° 

6045 

6063 

35° 

6065 

4991 

6° 

6045 

6053 

36° 

6066 

4930 

7° 

6046 

6041 

O'-'O 

6067 

4867 

8° 

6046 

6027 

38° 

6068 

4802 

9° 

6046 

6012 

39° 

6070 

4736 

10° 

6047 

5994 

40° 

6071 

4669 

11° 

6047 

5975 

41° 

6072 

4600 

12° 

6048 

5954 

42° 

6073 

4530 

13° 

6048 

5931 

43° 

6074 

4458 

14° 

6049 

5907 

44° 

6075 

4385 

15° 

6049 

5880 

45° 

6076 

4311 

16° 

6050 

5852 

6077 

4235 

1  t  ° 

6050 

5822 

47° 

6078 

4158 

18° 

6051 

5790 

48° 

6079 

4080 

19° 

6052 

5757 

49° 

6080 

4001 

20° 

6052 

5721 

50° 

6081 

3920 

21° 

6053 

5684 

51° 

6082 

3838 

22° 

6(54 

5646 

52° 

6084 

3755 

23° 

6054 

5605 

53° 

6085 

3671 

24° 

6055 

5563 

54° 

6086 

3586 

25° 

6056 

5519 

55° 

6087 

3499 

26° 

6057 

5474 

6088 

3413 

27° 

6058 

5427 

57° 

6089 

3323 

28° 

6059 

5378 

6090 

3233 

29° 

6060 

5327 

50° 

6091 

3142 

30° 

6061 

5275 

Gli°                    6092 

3051 

390 


TRIGONOMETRIC    FOBMl'  LAS. 


TABLE  XXVII.— TRIGONOMETRIC  AND  MISCELLANEOUS 
FORMULAS. 

TRIGONOMETRIC    FORMULAS. 


FIG.  98. 


In  Fig  99,  let  DCE  be  the  arc  of  a  quadrant,  ABC  a  right 
triangle,  the  angle  B AC  subtended  by  the  arc  CE  =  A,  and 
consider  the  radius  A  C  =  unity.  Then 


BC  =  si 
AB  =  cosA. 
HE  =.-  tan  A. 
DF  =cotA. 
AH =  sec  A. 


AF=  cosec^l. 
BE  =  versinJL 
DI  =  coversinvl. 


CF  =  coexsec^l. 


Using  the  small  letters  a,  5,  c,  to  represent  the  sides  of  a 
right  triangle  in  Fig.  98  or  99,  we  may  write 

sin  A  =  ® ;  cosec-4  =  -  ;  .-.  sin  A  = 

b  a  cosec^L 


c  b 

cosA  —  -;       sec A=-;    .-. 

b  c, 


sec  A 


tan  A  =  -]       cot  A  =  -  ;  .-.  tan  A  =  — 

c  a  cotA 


SOLUTION    OF    TRIANGLES. 


391 


TABLE  XXVII.— TRIGONOMETRIC  AND  MISCELLANEOUS 
FORMULAS. 

SOLUTION   OF  RIGHT  TRIANGLES. 


Required. 
A,  C,  C 

yl,  C,  b 

C,  b,  c 


CY,  a,  6 


Given. 


a,  6 


,  a 


Formulas. 


in  A  =  cos  C  =  - ;    c  = 


a)  (6  —  o)o 


tan  ^4  =  cot  7>  =  -  ;    b  =  */a?  +  c'2. 
c 


C  =  90°  —  yl  ;  a  =  b  sin  .4  ;  c  =  6  cosin  -4. 


SOLUTION   OF  OBLIQUE  TRIANGLES. 


Required. 


Area 

Area 

Area 


Given. 


A,   /,',  (I 


A,  a,  b 


«,  6,  c 


Formulas. 


a  sin  7) 


sin  7>  = 


.}(.!  +  7?)  =  £(180—  C) 

tan  £  (A  -  B)  =  ^^  tan  £  (/i  + 
a  +  b 

A  =  ±(A  +  77)  +  H^  —-?>*) 
7i=£(,4  +  B)  —$(A  -  B) 


If  s= 


™"H 


s  (s  —  a) 
he 


sin  ,1  = 


&c 


Area  =  v«  (.s  —  a)  (.s  —  6)  (.s  —  c) 

Area  =  1  hr  sin  /I 

-  sin  /I  sin  B 


Area 


2  sin  (      +  B) 


392  GENERAL   FORMULAS. 

TABLE  XXVII.—  TRIGONOMETRIC  AND  MISCELLANEOUS 
FORMULAS. 

GENERAL  FORMULAS. 

sinyi  =-v  1  —  cos2  .4  =tanylcos-4. 
sin  A  =  2sin^A  cos^A. 


sin  A  =  -  -  -  =  N/i(l  -cos^t). 
cosec  A 


—  sin2  ^4  =  cot  ^4  sin^l. 


-  — 
sec  .A 

=  1—2  sin2-J-^4  =1  —  versyl. 
cos,  A  =  *f 


tan  A  — 


cos^L 

sin  2  ^4 


cos  ^4  1  +  cos2  A 


cot  A          sin  2  A 


cot  A  =  —  -_  =  = 

tan  A       sin  yl 


1  —  cos2^4          sin  2  A 

sec  A  =  -  =the  reciprocal  of  any  expression  for  cos  A. 

cos  A 

cosec  A  =  -  =  the  reciprocal  of  any  expression  for  sin  ^4. 


vers  A  =  1  —  cos  A  =  2  sin2  •}  A. 


cos  A 


sin^-J1"^: 


GENERAL   FORMULAS.  393 

TABLE  XXVII.— TRIGONOMETRIC  AND   MISCELLANEOUS 
FORMULAS. 


xm^     1  —  cosA smA 

1  +  sec^l  sinvl  1  +  cos -4 

,    , 1  +  cosyl  _       sin/1 

sin^l  1  —  cos  .4 

sin  2  A  —  2  sin  A  cos  A. 
cos  2  A  =  cos2yl  —  sin2  A  =  2  cos2^!  —  1. 
2tanJ. 


tan  2A  = 


cot2A  = 


1  —  tan2 -4 
cot2  A  —  1 


2  cot  A 

sin  (^1  ±  72)  =  sin  A  cos  B  ±  cos^l  sin  5. 
cos  (A  ±  J>')  =  cos  .J.  cos  J>  qp  sin^l  sin/?. 
tan  >i±  tang 


1  if  tan  A  tan  7^ 
sin  A  +  sin  7>  =  2  sin  £  (  ^L  + 
sin/1  —  sin7>)  =  2cosi(yl  +  B)  sin  4  (A  —  B). 
cos  A  +  cos  7?  =2  cos  $(A  +  1?)  cosi(yl  —  B}. 
cos  73  —  cos  A  =  2sini(^L  +  72)  sin|(.l  —  B). 
sin2  A  —  sin2  B  =  cos2  B  —  cos^A  =  sin  (A  +  75)  sin  (A  —  B), 
cos*  A  —  s\ri2B  =  cos  (.4  +  B)  cos  (A  —  B). 


cos/1  cos  75 


. 
sinJS 


394 


MISCELLANEOUS   FORMULAS. 


TABLE  XXVII.— TRIGONOMETRIC  AND   MISCELLANEOUS 
FORMULAS. 

MISCELLANEOUS   FORMULAS. 


Required. 

Given. 

Formulas. 

Area  of 

Parallel  sides  =  in  and  n 

P  / 

Trapezoid 

Perp.  dist.  bet.  them  =  p 

|  (m  +  n) 

Regular  Polygon 

Length  of  side  =  I 
Number  of  sides  =  n 

nl*     ,180° 
—  cot  — 
4           n 

Circle 

Radius  =  r 

TZT2  [7T  =  3.1^ 

Ellipse 

Semi-axes  —  a  and  b 

Ttab 

Parabola 

Base  =  fr,  height  =  h 

nil 

Surface  of 

Radius  of  base  =  r 

Cone 

Slant  height  =  s 

nrs 

Cylinder 

Radius  =  r,  height  =  h 

2irrh 

Sphere 

Radius  =  r 

47Cf2 

Height  =  h 

i. 

Zone 

Radius  of  its  sphere  =  r 

Volume  of 
Prism  or  cylinder 

Area  of  base  =  b 
Height  =  h 

bh 

Pyramid  or  cone 

Area  of  base  =  b 

Height  =  JL 

Wi 
3 

Frustum  of 
Pyramid  or  cone 

Area  of  bases  =  b  and  &' 
Height  =  h 

:} 

Sphere 

Radius  =  r 

|7zrr3 

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